Translation:The zoo in my city has a hawk.
Yes, that is a tricky one. As a bird watcher I have a list of birds giving, Latin,Greek and English translations. Γεράκι according to this is used as a generic name for lots of different birds of prey. I don’t have the list with me here but from memory I think you are probably right. In the village they use diminutives for the smaller birds of prey and γεράκι for a long legged buzzed who nests in the valley.
Somewhere here is a discussion we had about how (most) Americans can't distinguish a rabbit (κουνέλι) from a hare (λαγός), so call them all rabbits. Often a hare is a jackrabbit. But i think hares are more common in Europe, so people know the difference (I had the same problem while living in Denmark, where the Easter Bunny is Påskehare.) But here I noticed that frogs and toads are all βάτραχος, which according to my dictionary is a frog, while a toad is φρύνος!