돌아오길 is colloquial contraction form of 돌아오기를 돌아오기를 is composed of 돌아오(다) + 기 + 를 돌아오다 means "return, come back" 기 is a suffix used for making a verb into a noun 를 is an object particle 부탁해 is contraction form of 부탁하여 and composed of 부탁하(다)+여 부탁하다 means "ask, request" 여 is an ending particle used after ~하(다) 돌아오기를 부탁해 means "request coming back" "어떻게 너 다시 돌아오길 부탁해?" How I request your coming back again?

돌아오다 come back, return 부탁하다 request, ask for 돌아오길 부탁해 means requesting someone to come back.