lfr, these are all indeed older, outdated English but very real spellings for these words, rather than simply being an invention of the author. "Poetick" and "philosophick" were out of date by the 1930s, but they are not his own creation.

You can find other examples of the "final k" example in words like "magick" and "musick". This was something that had disappeared from standard written English by the early 19th century. I know that "cooperation" with the diaresis was used by Shakespeare and I have seen it in books published as recently as the late 20th century (as well as in some newspapers that maintain this now-curious spelling in their official style guides).

I think that if it is Lovecraft he may have been trying to deliberately evoke an aura of "oldness" with the "final k" words rather than being joking. For "cooperation" he was just using what was then a common spelling