Is there a generic term for Lovecraftian horror that doesn't use the words "cosmic" or "horror"?

"Weird fiction" or "weird tales" are the terms Lovecraft used in Supernatural Horror in Literature. It's commonly associated with him and his contemporaries, and some modern authors (China Mieville, Jeff Vandermeer, among others) have been described as "New Weird". They usually lack conventional horror monsters and focus more on feelings of dread and the unknown, or the breakdown of reality. Wikipedia features a nice long list of authors if you want to see if that fits with what you're looking for.

The phrase Eldritch Abomination is associated with Lovecraft, perhaps you would prefer that to the word 'eldritch' on its own. Eldritch Abomination has an entry on TV Tropes, although I don't know how widely it is used.

From TV Tropes:

Native to H.P. Lovecraft's Cosmic Horror Story genre, the Eldritch Abomination has become a mainstay of horror and fantasy works, along with numerous others that derive inspiration from Lovecraft.