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According to Wikipedia

American writers Robert W. Chambers and H.P. Lovecraft borrowed the
term Carcosa for their stories.

I have copies of books that between them claim to be the complete stories and poetry of Lovecraft. I can find no mention of Carcosa in them. Did Lovecraft really incorporate Carcosa into his mythos?

He did not, at least not in his works meant for publication. (His letters might be another matter.) There is no mention of Carcosa in any of his prose fiction or poetry. His works (including some juvenilia and incomplete works) are all collected online here. It is a simple matter to see that "Carcosa" does not appear anywhere in the collection.

I have a vague impression of having come across some other writer(s) from Lovecraft's circle (perhaps Clark Ashton Smith) mentioning Carcosa in passing, but I cannot recall any specific instances.

The Whisperer in Darkness. It mentions the Lake of Hali, the Yellow Sign, and Hastur.

I found myself faced by names and terms that I had heard elsewhere in the most hideous of connexions—Yuggoth, Great Cthulhu, Tsathoggua, Yog-Sothoth, R’lyeh, Nyarlathotep, Azathoth, Hastur, Yian, Leng, the Lake of Hali, Bethmoora, the Yellow Sign, L’mur-Kathulos, Bran, and the Magnum Innominandum—and was drawn back through nameless aeons and inconceivable dimensions to worlds of elder, outer entity at which the crazed author of the Necronomicon had only guessed in the vaguest way. I was told of the pits of primal life, and of the streams that had trickled down therefrom; and finally, of the tiny rivulet from one of those streams which had become entangled with the destinies of our own earth.

Hali in Bierce's An Inhabitant of Carcosa:

Pondering these words of Hali (whom God rest) and questioning their full meaning, as one who, having an intimation, yet doubts if there be not something behind, other than that which he has discerned, I noted not whither I had strayed until a sudden chill wind striking my face revived in me a sense of my surroundings.

Hastur in Bierce's Haïta the Shepherd:

From this -- for he must be thinking if he would not turn into one of his own sheep -- he drew the solemn inference that happiness may come if not sought, but if looked for will never be seen; for next to the favour of Hastur, who never disclosed himself, Haita most valued the friendly interest of his neighbours, the shy immortals of the wood and stream.

The Yellow Sign in Chambers' The King in Yellow:

We talked on, unmindful of the gathering shadows, and she was begging me to throw away the clasp of black onyx quaintly inlaid with what we now knew to be the Yellow Sign.

Lovecraft apparently never mentioned Carcosa.

A search of this GitHub archive of Lovecraft's works turned up no results for "Carcosa." (To my additional surprise, I wasn't able to find any reference to a "King in Yellow" either!)

It's possible that this GitHub corpus could be considered incomplete; it doesn't include the stories he contributed to in an editorial or ghostwriterly capacity, and there's always the chance that the compiler missed something. It's hard to prove a negative, but no evidence in the positive is making itself known to me.

The only explicit link between Lovecraft and An Inhabitant of Carcosa seems to be via The King in Yellow. A list of eldritch entities in The Whisperer in Darkness includes "Hastur" and "the Yellow Sign," both of which show up in The King in Yellow first.