This is basically a question for literary studies as much as history, and there's going to be a degree of subjectivity in the interpretation of Lovecraft's beliefs and how they do and do not affect his fiction. It is entirely possible for an individual to believe one thing privately and write another thing—for example, Lovecraft was very much an ardent materialist in his own life, but much of his fiction involves sorcery and the supernatural.

So when scholars talk about the influence of Lovecraft's beliefs and philosophy having an affect on his writings, it usually means tracing back some specific instance or trend in his fiction to an aspect of his non-fiction writings—be that in his voluminous letters or his essays. The default work on the subject is S. T. Joshi's H. P. Lovecraft: The Decline of the West.

Joshi breaks Lovecraft's philosophy down into four basic areas: metaphysics, ethics, aesthetics, and politics; then examines how (and if) these areas apply to his fiction. Politics, for example, are not an aspect we normally think of when we read Lovecraftian fiction, but Joshi remarks on Lovecraft's support of fascism as expressed in his fiction (129-130):

Here the idea concerns surprisingly frequent attempts by authorities or other individuals to suppress information or to take the law into their own hands, for the supposed betterment or salvation of the human race. The notion sounds laudable, but raises troubling ethical and political questions: Who has appointed these figures to look after our welfare? How do we know that their efforts will be successful, or that the cure may not be worse than the disease? Is general ignorance of the angers facing us really better than knowledge, however psychologically devastating it may be? Lovecraft does not see to be bothered by these questions: he finds it self-evident that the intelligent and prominent will look after their lessers—occasionally at great sacrifice or risk to themselves—and rid the world of horrors it is better off not knowing.

This is an interpretation which is based directly off of Lovecraft's political thought, which was heavily class-based and supported the idea of intrinsic altruism among the most intellectually capable and sensitive individuals; Lovecraft's ideas on fascism are a bit too complicated to go into here, but are tied intimately into his classism.

In Lovecraft's nonfiction, pretty much all of which is contained in the five-volume set of his Collected Essays (mostly essays published in amateur journalism papers, letters to newspapers, etc.), Lovecraft's beliefs and philosophies are much more apparent, direct, and topical—addressing subjects of the day, such as the first World War, the Great Depression, or the Japanese invasion of Manchuria and their attendant philosophical aspects—questions on the nature of war, wealth, military service, etc.—and comparison with his letters shows that Lovecraft's views on these and other subjects shifted over time, but he was very consistent in accurately depicting his beliefs in his non-fiction essays.