かわいい is sometimes spelled in kanji as 可愛い. This seems to be an 音読み reading which points to a Chinese loan, and Chinese does have this word.
However, the meaning of 可愛 as "cute" in Chinese seems to have been a borrowing from Japanese. Originally it meant what its individual kanji meant, "able/deserved to be loved". For example, the Chinese Union Version of the Bible, translated in 1919, often uses 可愛 to mean "deserving of love", which is incongruent with its modern meaning borrowed from Japanese.
Also, かわいい seems to have been かはいい in 歴史的仮名遣い. はい would have come from "pai" which would be a stretch from on'yomi.
So 可愛い seems to be 和語. Why then did it get assigned a pseudo-音読み reading?