It's hard to answer the question of how someone felt inside, but to the extent we can judge a man's feelings and thoughts based on his words, the answer is no, he didn't. Or, at least, he didn't state in a public way that he did.
S.T. Joshi, certainly one of Lovecraft's most devoted historians, biographers, and, it must be said, fans, admits in his biography A Dreamer and a Visionary: H.P. Lovecraft in His Time that not only were Lovecraft's racism, anti-Semitism, ethnocentrism, and snobbery atypically vitriolic for his own time, but that Lovecraft failed to keep step with the generality of America as it moved away from said vitriol about race and culture:
"There is no denying the reality of Lovecraft's racism, nor can it merely be passed off as "typical of his time," for it appears that Lovecraft expressed his views more pronouncedly (although usually not for publication) than many others of his era. It is also foolish to deny that racism enters into his fiction."
Joshi goes on to recount an episode of correspondence between Lovecraft and a Mr. Charles D. Isaacson, in 1915, in which Lovecraft attacked Isaacson for espousing views of racial tolerance and for Isaacson's attacking the new film Birth of a Nation for stoking racial violence (Joshi, p. 97-99). Isaacson wrote back, publicly attacking Lovecraft for "[being] against tolerance of color, creed and equality, uphold[ing] race prejudice…"
Clearly, even in 1915, people's attitudes were shifting, but Lovecraft was doubling down on his viewpoints, even regressing on some. Indeed, his private writings, of which there are many thousands of letters over the course of his life, are virtually universally negative on non-WASP white ethnic groups such as German immigrants, Irish, and Catholics when he discusses them at all.
Part of this, it must be said (and Joshi points out), was almost certainly wrapped up in Lovecraft's Anglophilia and the 18th-century mannerisms and character which he affected for himself. As Lovecraft's friend Wilfred Branch Talman noted: it was only natural that a poseur of an 18th-Century English aristocrat should affect an attitude of racism and ethnic superiority, after all (Callaghan, p. 8). And because these attitudes were wrapped up very much in the way Lovecraft saw himself and the way he wanted to portray himself to the world -- and, indeed, in his writing, which is festooned with Anglicisms and archaisms -- he never really let go of them, even in later life.
Now, Joshi does argue that Lovecraft's viewpoints were more ethnocentric (or, to use a contemporary term, "nativist") than outright racist, as Lovecraft did sometimes express tolerance, or even praise, for groups he felt were "well assimilated." (Joshi, A Subtler Magick: The Writings and Philosophy of H. P. Lovecraft. pp. 107-108). Indeed, Lovecraft married a Jewish woman, Sonia Greene, who had emigrated to the U.S. as a child from Russia. So he was, at least, willing to make exceptions to his rules when he felt the candidate was culturally assimilated enough -- or when it suited him. It is worth noting, perhaps, that the marriage did not last.
To sum up, there is nothing in Lovecraft's letters or in his works of fiction -- which consistently demonstrate racist tendencies, such as portraying African natives as pseudo-beast-men -- to show that his attitudes toward race tempered or that he came to regret them, despite the changes in general American attitudes going on around him. He died at only 46, so he may not have had time to reach the age of reflection. Or, perhaps he simply held the beliefs too strongly for various personal reasons.
Sources
Callaghan, Gavin: H.P. Lovecraft's Dark Arcadia: The Satire, Symbology and Contradiction (2013)
Joshi, S.T.: *A Subtler Magick: The Writings and Philosophy of H. P. Lovecraft. (1996); A Dreamer and a Visionary: H.P. Lovecraft in His Time. (2001)
Lovecraft's views on race did not remain static throughout his life, but they changed very gradually and very little. He was never at any point "not racist" by contemporary standards, but for example, early in his life he was relatively biased against the Irish:
I regard the Celts as an inferior race, but little better than Mexicans, & but little more capable of self government. They could never maintain an orderly existence save under the domination of some branch of the Teutonic master race—if they could leave England, they would have to take Germany as a master; in fact, I am not sure but that they need a few Prussian methods to curb their ebullient & seditious emotions.
But this view changed over time, perhaps because of greater exposure to people of Irish descent (like his correspondent Robert E. Howard), perhaps because Lovecraft discovered he himself had a few Celtic ancestors, and perhaps because of the pointed arguments with his friends. In much the same way, several of Lovecraft's beliefs about race underwent slight changes over the course of his life - although never a real reversal. His arguments with friends like James F. Morton (an early member of the NAACP and author of a tract against race prejudice) was a particularly fierce opponent of Lovecraft's racial prejudices, and early positive view of the second Ku Klux Klan (Lovecraft's mentions of the KKK drop off after the scandals in the 20s).
Towards the end of his life especially, Lovecraft became more focused on cultural unity and continuity than biological racialism; one his more famous statements in that regard:
With the high-grade alien races we can adopt a policy of flexible common-sense—discouraging mixture whenever we can, but not clamping down the bars so ruthlessly against every individual of slightly mixed ancestry. As a matter of fact, most of the psychological race-differences which strike us so prominently are cultural rather than biological. If one could take a Japanese infant, alter his features to the Anglo-Saxon type through plastic surgery, & place him with an American family in Boston for rearing—without stemming him that he is not an American—the chances are that in 20 years the result would be a typical American youth with very few instincts to distinguish him from his pure Nordic college-mates. The same is true of other superior alien races including the Jew—although the Nazis persist in acting on a false biological conception.
Lovecraft tended to wear his prejudices on his sleeve, and in trying to justify some of those prejudices he supported certain courses of scientific racialism, and denied newer evidence that suggested otherwise; this is rather against his approach with other subjects, like physics, where Lovecraft was more accepting of new ideas and information. Case in point:
What I was really laughing at was no Boas himself—whom I freely gave a place among the first-rate anthropologists—but the naïve way in which all nigger-lovers turn to him first of all when trying to scrape up a background of scientific support. He is the only first rate living anthropologist to overlook the obvious primitiveness of the negro & the australoid, hence the equalitarian Utopians have to play him up for all he’s worth & forget the great bulk of outstanding European opinion—Boule, G. Elliot Smith, Sir Arthur Keith, &c.
In this case Lovecraft is poo-pooing Franz Boas, an anthropologist who worked to disprove the idea of fixed biological races; the others HPL mentions were proponents of scientific racialism. The particular fixation of black people (including native Australians, whom Lovecraft never met) as "inferior" biologically and culturally was a very strong prejudice he held from a young age, and continued to express until his death. Lovecraft never used the term "great chain of being" but his letters make it clear he did tend to think of evolution as being from "simple to complex" or "less advanced" to "more advanced," and postulated that black people had evolved possibly separately and more recently than other "races."
Lovecraft did not express much regret for his views towards the end of his life, or at least there is no regret expressed in his letters or in the surviving memoirs of his wife and friends. That doesn't mean he might not have been chagrined about some of his earlier views and letters, but he didn't express it as such. He did take pains to try not to express his views in such a way as to offend any of his friends. For example, in the margin of one letter he wrote:
P.S. Better not quote any of this to Bloch (who I discover is of Jewish extraction). While of course this question does not involve any aspersion on the Jewish heritage as a whole, it nevertheless makes embarrassing reading for anybody having more than an academic connexion with Semitism. One would handle it differently with a Jewish correspondent.
Lovecraft's views on Jewish peoples is another that underwent a change over the course of his life, although perhaps too much to go into exact detail here; he was never not anti-Semitic by contemporary standards, but his individual views and arguments shifted over the course of his life - both positively and negatively - but he did become much more keen not to offend folks with those views later in life.
Very interesting read thanks