Can't I say "무엇을 아이에게 가리칠 것이에요?"
You can. "무엇을 아이에게 가르칠 것이에요?" but I think it should be "아이들", not "아이" because of "the children"
If you were to do it without the next syllable I think it would be the child
아이한테 뭘 가르칠 거에요? Will this work too?