I see lots of birds with injuries or diseases that are quite disturbing, but with no way to treat them in the wild, they continue on as best they can. My guess is this poor fellow got too close to a hunter with a shotgun, a beautiful bird with a big chunk of his beak missing.
Finches are quite susceptible to Avian Pox which inflicts ugly tumors on their feet and around their eyes. I remember vividly, watching a handsome House Finch that entranced me totally until he turned his head. Most of the opposite side of his face was missing. I was glad that he was at our feeder and didn’t have to compete for his food. A year or so ago we had an outbreak of Salmonella amongst our Pine Siskins that killed a lot of birds. The grim side of birding.