The term "Celtic" has historically been used as an imprecise catch-all that vaguely applies to that which is European, non-English and "other".
Kernow is in fact the name for the geographic region the English call "Cornwall", which means "land of Cornish foreigners" in Old English.
There are in fact at least three different revived "Cornish" languages including Unified-Cornish, Phenemic-Cornish and Traditional-Cornish.
Cornish is better referred to as "Karnewek", "Kernewek" or "Curnoak". But if you're writing in English, why not just say "Cornish"?