There is a pretty long quote from Peter Kenez' "The Ideology of the White Movement" that sums it up quite well:
"Was the Volunteer Army conservative, reactionary, or should it be described as fascist? In the context of Russian counter- revolution let us term conservatives those who accepted the tsarist political and social system as basically just and suitable for Russia, but at the same time were willing to accommodate themselves to the changes which were occurring in that society. The reactionaries were those who posited a past utopia and criticized tsarist society because it had abandoned some of the institutions and principles which in their opinion had made the country great. The reactionaries, unlike the conservatives, were attracted to radical solutions. We can consider as fascist those reactionaries who by demagogic means attempted to build a large-scale popular movement. Fascists have struggled with particular bitterness against the emancipatory trends of Western civilization and therefore their criticism cuts across national boundaries. [...]. Generals Alekseev and Denikin, the two men who did most to develop a political programme for the Volunteer Army were conservatives. Compared with the vast majority of their followers, they possessed a degree of political sophistication and tolerance. Consequently, if we form a judgement on the basis of the public statements of the Army, we must conclude that it was a conservative organization. However, the officers who joined the movement in the beginning of 1918 and continued to play a dominant role in it until its final defeat, were young, profoundly embittered by their recent experiences in the Revolution and drawn to radical solutions. Especially after Alekseev's death in the autumn of 1918, Denikin was constantly under attack from the right; his followers wanted him to pursue reactionary policies. In April 1920 Denikin had to go into exile and his successor, General Wrangel, was a candidate of the right. To be sure, Denikin lost out in the power struggle largely because his armies had suffered defeats; nevertheless it is clear that his position had been greatly weakened by political disagreements. If we judge by the sentiments of the majority of the officers, and on the basis of the policies which the peasants actually experienced at the hands of the Volunteer Army, we should consider the White movement a reactionary phenomenon. The majority of the peasantry was right to distrust the promises and public statements of the Whites."
In their speeches and manifestos, the White leaders continuously emphasized that the Army fought for popular sovereignty – that only the Russian people could decide over the important questions of the day (the status of the monarchy, the question of land reform, etc.) through an elected Constituent Assembly. To quote the Volunteer Army's first public statement, the manifesto of January 9th 1918:
"The new Army will defend civil liberties in order to enable the master of the Russian land – the Russian people – to express through the elected Constituent Assembly its sovereign will. All classes, parties and groups of the population must accept that will. The Army and those taking part in its formation will absolutely submit to the legal power appointed by the Constituent Assembly".
After Kornilov's death, Denikin continued to uphold the former's line in regards to the goals of the Army. To quote Peter Kenez' "The Ideology of the White Movement":
"Extreme rightist groups in South Russia organized armies with German help and denounced the Volunteers as 'having an undetermined political physiognomy' and being 'covert republicans'. This monarchist propaganda was powerful, and many officers who were disenchanted with Denikin's liberalism joined the rightist armies instead. Denikin, however, did not give in. In May 1918, he addressed his small army on this subject and explained to the soldiers that raising the monarchist flag would be wrong, for the Army could not decide for the people. The task of the Army was to defeat the Bolshevik enemy, and monarchism would make this task more difficult by alienating moderate support. Undoubtedly Denikin was correct in his evaluation."
The same views are expressed in Denikin's and Alekseev's joint manifesto from May 1918, called "Goals of the Army”. It said the following:
"I The Volunteer Army is fighting to save Russia by (1) creating a strong, disciplined and patriotic army; (2) waging merciless war against Bolshevism; (3) reestablishing unity and lawful order in the country.
II In attempting to cooperate with all politically-minded Russian people [gosudar stvenno my sliashchimi], the Volunteer Army cannot take on a party label.
III The question of what form the state is to take should be postponed to the last stage [of the fight] and will be the expression of the will of the Russian people after it is freed from tyranny and madness.
IV [The Volunteer Army will have] no contact either with the Germans or with the Bolsheviks. To us only these positions are acceptable: the Germans must withdraw from the territory of Russia; the Bolsheviks must capitulate and be disarmed."
However, despite all their talk about popular sovereignty and the Constituent Assembly, most of the White leadership staunchly refused to recognize the original Constituent Assembly elected in 1917 as legitimate and demanded new elections to be held after the victory over bolshevism. To quote Kenez' "The Ideology of the White Movement":
"During the early months of 1918, there was a general consensus that only an elected Constituent Assembly could decide over the future of Russia. At this time the only issue was whether there should be new elections after the defeat of the Bolsheviks. The anti-Bolshevik socialists, for understandable reasons, insisted that the elections had already taken place and therefore the goal of the struggle could only be to restore the right of the duly elected representatives to make their decision. The non-socialist politicians and the leaders of the Volunteer Army, on the contrary, equally understandably, wanted new elections, arguing that the results of the elections no longer reflected the will of the people. The dispute, which was divisive and bitter, was narrowly political and did not concern a matter of principle."
The officers, which from it's early beginnings untill it's eventual defeat constituted the core of the White Movement, loathed not only the Bolsheviks, but all self-described socialists. To quote Peter Kenez' "The Ideology of the White Movement":
"The officers felt misunderstood, and blamed above all socialist and pacifist propaganda. Not surprisingly, they had a special animus against the great novelist, L. N. Tolstoy, an ex-officer whom they considered a traitor and renegade. The notion of non-resistence to evil seemed mischievous. The officers retorted to Tolstoy that killing in war was not like simple murder because soldiers killed without hatred. They imagined pacifists and socialists to be allies. There is an amusing scene in the dreadful novel by General P. N. Krasnov, From Double Eagle to Red Flag, in which the revolutionaries cheerfully discuss what a good job Tolstoy is doing in undermining the Army. The officers rejected with scorn the socialist proposal to transform the army into a people's militia. In general, they condemned socialism as materialistic and anti-individualistic in contrast to 'spiritual' militaristic values."
And to quote Admiral Kolchak himself:
"The general opinion...was that only a government authorized by the Constituent Assembly could be a real one; but the Constituent Assembly which we got...and which from the very beginning started in by singing the 'Internationale' under Chernov's leadership, provoked an unfriendly attitude...It was considered to have been an artificial and a partisan assembly. Such was also my opinion. I believed that even though the Bolsheviks had few worthy traits, by dispersing the Constituent Assembly they performed a service and this act should be counted to their credit."
Alekseev even went as far as to state that he would rather cooperate with Lenin than with Savinkov and Kerensky.
The politicians of the White Movement had learned the technique of avoiding controversial issues in 1917 under the guise of respect for the Constituent Assembly. And while people like Kornilov and Denikin might've actually believed in the principle of popular sovereignty (or rather their distorted understadning of it), the majority of the White officers couldn't have cared less about "politics" and the Constituent Assembly. However the more intelligent among them realized that for two different reasons it was a necessary compromise during the times of civil war. On one hand, the espousal of any specific form of government would've certainly alienated potential supporters, and on the other hand paying lip service to popular sovereignty was neccessary in order to obtain aid from the Allies. As one Russian journalist observed at the time:
"In Omsk itself ... could be seen a political grouping who were prepared to promise anything that the Allies wanted whilst saying that 'When we reach Moscow we can talk to them in a different tone' ".
Thus the White's political program was not so much a reflection of the actual beliefs of it's followers or even it's leaders, but a carefully constructed compromise that would've probably broken down once the civil war was over.
Now, when it comes to the actual beliefs of the Whites, the officer corps and even the ethnic Russian rank-and-file were overwhelmingly monarchist. To quote Peter Kenez' "Red Attack, White Resistance: Civil War in South Russia, 1918":
"From its inception, the army attracted a large number of rightist extremists and reactionaries. The experiences of the Ice March, in the course of which the men almost always lived in a hostile environment, created much bitterness, which is always a potent source of extremism. In the early summer of 1918, it was estimated that 80 to 90 per cent of the non-Cossack segment of the army was monarchist. The monarchists formed secret societies and began to display tsarist medals and flags. Denikin faced great pressure to come out for monarchism, not only from his soldiers but also from such popular commanders as Drozdovskii, who belonged to one of the secret societies."
And to quote Peter Kenez' "The Ideology of the White Movement":
"After assuming command in April 1918, Denikin found himself increasingly under monarchist pressure. The first few months of Soviet rule pushed non-Bolshevik public opinion in a conservative direction. Monarchists, though they were a small minority, started to organize. From the beginning the overwhelming majority of the officers serving in the army were monarchists. Denikin, who inherited the leadership of the Army as something of an accident, had neither Kornilov's charisma nor his prestige within the movement and consequently found it hard to resist the pressure. An early hero of the White movement, Colonel Drozdov- sky, openly advocated the restoration of the Romanovs. Denikin's closest advisers, Generals A. S. Lukomsky and A. M. Dragomirov took the same view, and the increasingly ailing General Alekseev hesitated. Extreme rightist groups in South Russia organized armies with German help and denounced the Volunteers as 'having an undetermined political physiognomy' and being 'covert republicans'. This monarchist pro- paganda was powerful, and many officers who were disenchanted with Denikin's liberalism joined the rightist armies instead. [...]. Allied victory in Europe, which promised increased foreign aid, greatly improved the chances of the Whites in the Civil War, and this development, too, encouraged the monarchists. They believed that the time had come when they did not have to make compromises any more. When Generals Lukomsky and Dragomirov told Denikin in the autumn of 1918 that they could no longer serve in an army whose primary goal was the restoration of the Constituent Assembly, the Commander-in-Chief modified his position. From that time on the statements of the Volunteer Army mentioned not a Consituent Assembly, but a National Assembly. It is hard to say exactly what was the significance of the change in ter- minology and Denikin's public statements in the course of 1919 did not clarify the matter. The anti-Bolshevik socialists suspected, probably cor- rectly, that the change meant that the functions of the envisaged Assembly would be somehow limited and that the Volunteer Army no longer advocated universal suffrage. Although the Whites continued to talk about the necessity of allowing the expression of popular will, their lack of precision and their vacillation in this matter make it impossible to say whether the Volunteer Army at the time of its greatest victories stood for popular sovereignty."
From late 1919 onwards, the Volunteer Army even began to purge the anti-Bolshevik self-described socialists. To quote Victor Bortnevski's "White Administration and White Terror (The Denikin Period)":
"In November 1919 the Special Council granted extraordinary authority to the cities to collect funds for reinforcing measures against robberies and to organize local self-defense units. The chiefs of uezda were empowered to organize special units (consisting of up to four hundred members) to fight against "banditism" and to form uezd state-security reserves, consisting of mounted and infantry units, to combat gangsterism. That same month the Special Council decided to increase the severity of punishment for offenses specified in the laws of 22 July 1919. The death penalty was henceforth imposed for "membership in the Bolshevik/Communist parties, Soviets of workers, soldiers and peasant deputies, or other similar organizations which participated in the grab for power by the Soviets, or persons who supported the policies of this power." Similarly, the death penalty was specified for ordinary members of the Bolshevik party, as well as for members of the Central Committee and the Sovnarkom. According to the letter of this law, members of the Socialist-Revolutionary, Menshevik and the People's Socialist parties were also subject to the death penalty since, as was well known, these parties had collaborated in the grab for power during the February Revolution. Most peculiar in this respect was the fact that some prominent members of these parties were loyal members of the Denikin administration; they were active in various political institutions, the press and the zemstvo and city councils in the White south."
Kolchak's regime didn't treat them much better either. To quote from Orlando Figes' "A People's Tragedy":
"For several weeks after the coup, Kolchak’s police carried out a series of bloody reprisals against SR activists. Hundreds were arrested — many as ‘hostages’ to be executed in the event of SR acts of terror against the dictatorship. Among the hostages in Omsk were twenty SR deputies of the Constituent Assembly, ten of whom were shot in December following a workers’ uprising in the town."
To honestly believe that the Whites, who literally shot any self-described socialist on the spot and used their Constituent Assembly delegates as hostages that were to be executed in the event of an uprising, would have allowed the free election and formation of a Constituent Assembly that would innevitably be dominated by self-described socialists (and at the time everyone with a functional brain knew that) is naïve at best.
So yeah, here we have it. On paper the Whites fought not for any specific form of government, but for "popular sovereignty" (i.e. an elected Constituent Assembly without any self-described socialists). However the vast majority of those who fought for the Whites, and especially it's it's officer corps, were overwhelmingly monarchist. I seriously doubt that, once the civil war is won, the officers would care at all about popular sovereignty (which they never believed in in the first place). At that point, what Denikin wanted would be of little importance. We saw in OTL how easily the officers could force their commander-in-chief to resign (although his OTL deposal was more the result of military failures).
The vast majority of the workers and peasants knew that. The Whites wore tsarist uniforms, chanted tsarist songs, carried tsarist banners and behaved like the old tsarist army had. White bataillons were led by former tsarist officers and the functionaries installed by the Whites often were former tsarist bureaucrats. The Whites returned the redistributed land to the old tsarist landowners, and punished anyone who had taken part in said redistribution.
And as for Great Russian chauvinism, even Denikin allways used the old tsarist term "Malorossia" (little Russia) when talking about Ukraine.