Even though this is a jukugo word, it doesn't use the on'yomi readings. Both the kanji 子 and 犬 often favor kun'yomi readings, and this word is no exception. The reading for 犬 is the one you already know, and the reading for 子 as こ is a really common one (perhaps you've already learned that reading?). If not, just think about punching a puppy in the face, and giving it a KO (That's a knock out).
The puppies were born this morning.
The puppy that was standing at the entrance was given the nine balls around its neck by the prince.
Let’s pretend we’re puppies!