What the hell is a “Satyr” in the Lovecraft world-view? Let me bloviate. In Lovecraftian tales, the human genotype is subject to degeneration or mutation under the influence of the Outside. Sometimes this is done purposefully, as with ghouls (who originate from humans), Deep Ones (who interbreed with us), or Yog-Sothoth (whose hybrid spawn also interact).
But there are other ways for humanity to become animalistic and vile – in Lovecraft’s tales you have the monstrous remains of quadruped humans in the Rats in the Walls, the horrible Martense family from the Lurking Fear, and Arthur Jermyn. Ramsey Campbell made this trend very clear in his own stories, particularly with The Moon Lens, in which a person is engulfed by what might be Shub-Niggurath, and then spat out as a terrifying monster, as well as the Horror Under Warrendown, in which the animalistic horrors renew their numbers by feeding their god’s “fruit” to others.
Basically my theory is that all of these deteriorated, genetic mutations in the various Mythos tales may have a thousand origins, but that in all their variety, as a group they are the Satyrs which are also mentioned in Mythos lore. So I present these hobbling, chuckling, mowing, and enthusiastic creatures as my carefully-presented analysis of the Lovecraftian Satyr.