Sortof.
WARNING: THIS ANSWER CONTAINS RACIST SPEECH IN LETTERS QUOTED FROM THE PERIOD
Genocide as a concept already existed in the 1920s and 30s, but in the decades before the Holocaust the idea of mass murder on that scale was not in the common parlance, and most people didn't seem to be able to conceive of it. When you look at the stories of Robert E. Howard, for example, there are races of people that go extinct, or are on the downward trend toward extinction, but it is usually a long period of slow decline that sometimes reaches a fearsome and suddenly bloody ending. Race war was seen by many as inevitable in the early 20th century, and Lovecraft was not immune to "future war" scenarios such as this:
Of Japan I have not so far spoken, because I think it a certain enemy of the future, which no plan can permanently make a friend. It demands free access to Anglo-Saxon soil for its citizens, and this can never be given. Orientals must be kept in their native East till the fall of the white race. Sooner or later a great Japanese war will take place, during which I think the virtual destruction of Japan will have to be effected in the interests of European safety. The more numerous Chinese are a menace of the still more distant future. They will probably be the exterminators of Caucasian civilisation, for their numbers are amazing. But that is all too far ahead for consideration today.
In general it can be said that Lovecraft did not advocate mass murder, either on a racial or other bias, although he was fully supportive of the United States and United Kingdom during WWI against the Germans, although given that he considered the nations involved all "Aryans" he considered it a terrible mistake. He wasn't completely apathetic with regard to massacres and atrocities committed against non-white people, and noted once:
Late in 1675 the Mass. rulers sent a body of mixed Mass., Plymouth, and Conn. troops into the Narragansett region without our permission and against our wishes, and on one terrible winter's night descended on the tribe's retreat in the swampy woods and massacred every living redskin they could find—men, women, and children alike. This is the "Great Swamp Fight" of history—the Lord's Brethren against the wicked pagan! This was virtually the end of the Narragansett nation.
In this case, Lovecraft is decrying the nigh-genocidal massacre of the Native Americans of Rhode Island (the royal "we" he uses above; Lovecraft was a Rhode Islander). Generally speaking, he wasn't an advocate of violence - and also generally speaking, didn't say much against violence when it was done, especially if by the government in the course of its duties or if it conformed to his prejudices (he rarely mentioned or spoke out against lynchings, for example.)
Yet the Great War had introduced new methods of mass murder: poison gas, submarines, airplanes, and artillery had massively increased in both scale and effectiveness to anything that had ever existed before, and while Lovecraft did not serve in the armed forces during World War I, he was aware of these technological developments...and it is in this sense that Lovecraft does make some really questionable statements in a few of his letters.
We walked—at my suggestion—in the middle of the street, for contact with the heterogenous sidewalk denizens, spilled out of their bulging brick kennels as if by a spawning beyond the capacity of the places, was not by any means to be sought. At times, though we struck peculiarly deserted areas—these swine have instinctive swarming movements, no doubt, which no ordinary biologist can fathom. Gawd knows what they are—Jew, Italian, separate or mixed, with possible touches of residual Irish and exotic hints of the Far East—a bastard mess of stewing mongrel flesh without intellect, repellent to eye, nose, and imagination—would to heaven a kindly gust of cyanogen could asphyxiate the whole gigantic abortion, end the misery, and clean out the place.
For context, Lovecraft had been invited down to New York for the first time in his life in 1922; he and his friends went to see all the sights, which involved several of the ethnic neighborhoods with their tenements overflowing with newly-minted New Yorkers from all over the world. Already a bit of a nativist, visiting a massive city for the first time (in 1920 New York City had 5.6 million people; Lovecraft's hometown of Providence less than half a million), and was overwhelmed by the number and diversity of the inhabitants.
The best that could be said is the statement wasn't serious; equivalent to a "first, kill all the lawyers" joke. The specific method, however, comes from the First World War, where a deadly cloud could roll across a battlefield and extirpate all life. This can be seen again in a later comment:
I’d like to see Hitler wipe Greater New York clean with poison gas—giving masks to the few remaining people of Aryan culture (even if of Semitic ancestry). The place needs fumigation & a fresh start. (If Harlem didn’t get any masks, I’d shed no tears ….. & the same goes for the dago slums!)
This is very specifically about New York. Since writing the first letter, Lovecraft had moved to New York, married, his marriage had failed, he had "failed to make it" in New York City, and returned to Providence to live with his aunts in genteel poverty - and for the rest of his life Lovecraft would hold a detestation of "the Pest Zone," with its large immigrant population. Adolf Hitler became Chancellor of Germany in January 1933, and his antisemitism was already grossly apparent, though the full scope of what would become the Holocaust was not apparent. Lovecraft was tentatively supportive of Hitler, who he thought was a clown, because he approved of German nationalism, though he thought the Nazi party's antisemitism was unscientific. This support quickly waned as the Nazis passed more and more laws restricting Jewish life; Lovecraft's prejudices and an inherent centrist approach prevented him from completely condemning Hitler or the Nazis. He would die in 1937 before the horrors of the Holocaust became publicly known in the United States.
Lovecraft's particular prejudice regarding New York was that the place had so many unassimilated immigrants that it was basically no longer a part of American culture, and there was no way to assimilate it as-is. In no small part this has to do with Lovecraft's antisemitism and conspiracy theories about the reach and influence of Jewish peoples in New York on the publishing industry and newspapers - I'll spare you his theories about "the Jew-York papers" - and again, it's hard to tell how much he's being serious versus engaging in a degree of hyperbole.
The idea does crop up at least one more time in his letters, this time with a slightly different target:
The Aryans in India were too late in establishing their colour-based caste system, so that today the culture of the Hindoo is probably the most thoroughly repulsive on our planet. The more one learns about India the more one wants to vomit. Aside from a few profound minds, the Indian people represent such an abyss of degeneracy that extirpation & fumigation would seem to be about the only way to make Hindoostan fit for decent people to inhabit.
Lovecraft never visited India, which at the time was still under British rule (although Gandhi & co were working on that!), and his second- and third-hand misunderstanding of the situation basically made it appalling to a white male American who believed firmly in the "color line" as practiced in the United States under Jim Crow. He spoke out of near-total ignorance and apathy, and along the same lines as in previous instances.
It's important to emphasize that Lovecraft wrote a lot of letters; these are a handful of instances take from the tens of thousands of letters he wrote, spread across a couple of decades. Taken out of context, they look bad - and the prejudices involved are bad - but they are not a call for action and should not be understood as such.