What would you call a baby hawk?

A baby hawk is an eyas. A falcon chick, especially one reared for falconry, still in its downy stage, is known as an eyas. Adult falcons have thin, tapered wings, which enable them to fly at high speed and change direction rapidly. Fledgling falcons, in their first year of flying, have longer flight feathers, which make their configuration more like that of a general-purpose bird such as a broadwing.

This characteristic makes flying easier while learning the exceptional skills required to be effective hunters as adults. The technique of hunting with trained captive birds of prey is known as falconry.