Avatar Press likes to use variant covers to promote its series, particularly short-run limited series like Moore’s; besides their artistic merit, several of these covers have more to say about the contents of the book than readers might think.
Note that, especially for many Dreamscape and Pantheon variant covers, the variant cover subjects depicted pertain to Lovecraft fiction, but not necessarily to the events depicted in Providence comics.
With the cephalopod-like head surmounted by a spider-like array of eyes and rudimentary wings or fins on the back, this may be intended to be a depiction of Cthulhu from “The Call of Cthulhu“:
If I say that my somewhat extravagant imagination yielded simultaneous pictures of an octopus, a dragon, and a human caricature, I shall not be unfaithful to the spirit of the thing. A pulpy, tentacled head surmounted a grotesque and scaly body with rudimentary wings; but it was the general outline of the whole which made it most shockingly frightful. Behind the figure was a vague suggestion of a Cyclopean architectural background.
Commenter Troy Leaman suggests:
#1 is definitely Hastur…at lake Hali.
Along the shore the cloud waves break,
The twin suns sink behind the lake,
The shadows lengthen
Strange is the night where black stars rise,
And strange moons circle through the skies,
The cover portrays the “Lilith” creature from Providence #2, based on the entity from “The Horror at Red Hook.” Notice also the three parallel claw-marks on the nearest rock, a reference to the creature’s passing in the issue.
The eponymous “Dagon” from Lovecraft’s story of the same name.
The globules or spheres suggest this is a representation of Yog-Sothoth, as described in “The Horror in the Museum”:
Imagination called up the shocking form of fabulous Yog-Sothoth—only a congeries of iridescent globes, yet stupendous in its malign suggestiveness.
A depiction of Shub-Niggurath, or more likely one of her Dark Young, fairly similiar to the Call of Cthulhu Roleplaying Game art for the same.
Probably intended to be Bokrug, the water-lizard deity from “The Doom that Came to Sarnath”; alternately, it could be a depiction of Clark Ashton Smith’s toad-like entity Tsathoggua, though it lacks that critter’s bat-like characteristics.
Possibly a depiction of Azathoth, the “Nuclear Chaos,” surrounded by its demon pipers.
Probably a bust of Hypnos, the Greek god of sleep, from Lovecraft’s story of the same name. (Thanks to commenters Blackpill, MS, and Lalartu for pointing this out.) Compare with this surviving bust of Hypnos from the British Museum:
This appears to be Yig the Father of Serpents who appears in “The Curse of Yig” by Lovecraft with Zealia Bishop.
The cover depicts Johnny Carcosa, who appears in Moore and Burrows earlier Lovecraft stories and comics The Courtyard and Neonomicon. See annotations for Neonomicon #1 P15,p4.
The cover depicts the High-Priest Not To Be Described from The Dream-Quest of Unknown Kadath.