This depends on what you mean by "his time." Lovecraft liked to frame himself as a traditional conservative of the 19th century. In fact, when he began as an amateur journalist his first major statement of philosophical intent came with the publication of his Newspaper 'The Conservative.' He declared his allegiance to:

“Total abstinence and prohibition; of moderate, healthy militarism as contrasted with dangerous and unpatriotic peace-preaching; of Pan-Saxonism, or the domination by the English and kindred races over the lesser divisions of mankind; and of constitutional or representative government, as opposed to the pernicious and contemptable false schemes of anarchy and socialism. Though the first-named of these items may superficially appear a rather inappropriate function for a Conservative, it must be remembered, that he who strives against the Hydra-monster Rum, strives most to conserve his fellow-men.”

He was a conservative in a very literal sense. He wasn't focused in the politics of his time, but on preserving the politics of an age which he felt was more dignified and safe.

“I really am a relic left over from Queen Anne’s age. I do not know how it came about, but from the time of my earliest recollection, I have seemed to fall into the mental habits of two centuries ago.”

However this was early in his public life (The Conservative was first published when Lovecraft was 25). Lovecraft actively pursued modern scientific and philosophical developments which aligned with his view of science being the only infallible form of analysis (In the early 20's he wrote that "right and wrong are primitive conceptions which cannot endure the test of cold science). He often paired his scientism with his thoughts on fascism, which he was drawn to. Democracy was a "false idol - a mere catchword and illusion of inferior classes, visionaries, and dying civilizations." In the same letter he wrote that since ethics were just mere prejudices and illusions... there is no earthly reason why the masses should not be kept down for the benefit of the strong."

His thoughts on race and fascism would align him with Hitler, whom Lovecraft supported at the end of his life. However two major shifts muddy the image of his connection to modern politics. One was the robbery of his house in May of 1925. Scholar Johannes Weyrauch noted that this infamous event caused a major shift in HPL's work. Before the 'inferior' races were individual villains who could be defeated or escaped from. After, they were an inescapable doom which surrounded respectable society and were inevitably going to destroy it. Even the hard right of the period thought that society could persevere. This development seems to be something largely new. On the other hand, Lovecraft responded to the difficulties of the 30's by championing modern values like unemployment insurance, and old age pensions. He became a mild socialist, while simultaneously believing only the strong members of society should have the vote. The "unstable nature of capitalism" needed to be checked. When he wrote of Hitler after this development, it had clearly affected his ideas of race.

"Hitler is an unscientific extremist in fancying that... even a trace of non-Nordic blood - or not Aryan blood - is enough to alter the psychology and citizenly potentialities of an individual. These assumptions, most certainly, are crude and ignorant...The races are equal but infinitely different... The effective development of civilization depends largely upon its stability and continuity; and these factors cannot be endured unless (a)the culture-stream remains relatively undiluted by alien traditions, and (b) the race-stock remains approximately the same as that which evolved the culture and institutions now existing."

Lovecraft here is reacting to both Hitler (whom he supports) and the "rabble-catering equalitarian columnists of the Jew-York papers." He is being influenced by the politics of his time by reacting against them.

Lovecraft begins his life being a staunch antiquarian, and ends it by reevaluating his views and re-framing his racial beliefs in contrast with the politics of his modern time. You can argue that he was not influenced at all by politics until his later socialism, but a more nuanced answer would be to say that the majority of his career was framed as opposition to modernity. Modern politics influenced him, but for a majority of his life they pushed him into staunch almost cartoonish conservatism. In his later life he attempted to merge his bigotry and fascist support with a belief in the need for socialism in the face of the great depression.

Sources:

Lord of a Visible World: An Autobiography in Letters by H.P. Lovecraft

Selected Letters Vol. 1-5 by H. P. Lovecraft

H. P. Lovecraft: A Life by S. T. Joshi

Racism and White Anxiety in Lovecraft's Weird Tales by Johannes Weyrauch

In his letters between 1929 and 1931 (so context: dealing with one of the biggest crises capitalism had faced up until then and with the appearance of an alternative to this system in the shape of the Soviet Union, all meanwhile fascism was popping up in Europe) he mentioned several times his conviction that advanced capitalist countries could not survive unless they evolved into absolutist dictatorships or reached some kind of wealth and income distribution. I don't recall him using the term "welfare state" but I think what he was proposing we would understand nowadays as that in the sense that he did not propose overcoming the capitalist mode of production and evolve towards another system, but more like "rich people sharing what they get or else workers are gonna blow everything up". Source: Selected Letters of H. P. Lovecraft III (1929–1931). Arkham House.