It's true, I quite like Pigeons
Getting a Hawk in around the breeding season is the tactic to make them go elsewhere
The problem with flying Hawks in the "Breeding Season", is there isn't one, it's all year round if conditions are favourable (temperature doesn't seem to matter, but they do respond to the availability of food). Falconers' Hawks do occasionally catch and eat sick Feral Pigeons, which isn't good news for the local wild Peregrines and Sparrowhawks.
I've also seen plenty of Gulls with missing feet/legs, Ducks and Geese too. Nylon Fishing line is often the culprit for birds in or near water, I guess (only a guess) Fishing Nets drying on beaches catch a few too. As @Nutcracker said above; rubbish at landfill sites maims/injures many.