Perhaps a sign would help with the eviction!
I have hawks here at most times of the year, although it tends to be the males that hunt the smaller birds at my feeding stations. They do indeed have a role to perform in ensuring the fitness of bird populations and a study recently showed that they are vital in protecting songbird nests.
Basically, small birds nesting within the hawk's territory are protected from corvids and don't become a menu choice until later in the year when the chicks fledge and need plenty of prey nearby. In this way the songbirds raise a brood or two with lower predation pressure. As the male does most of the hunting in the early days, smaller birds are targets but once the young are old enough to be left by the female she'll target pigeons and doves that last the family a bit longer.
In previous gardens I've used trellis to limit flight paths and enable small birds to escape but here I do nothing and enjoy the balance. Hawks don't catch much and last winter a one-eyed brambling was a regular for three weeks, despite two male hawks hunting the feeders. She disappeared with the other migrants.