The wounds, you have flushed them really well with saline or something like Chlorhexidine correct?
For his face and eyes, I would use saline to wash the blood out, taking breaks if he's stressed.
Plain Neosporin can be used in his injured eye as well.
I would use causing with the heat and wrapping the wound. The wound unless he's picking at it, personally I would leave unwrapped, but keep moist with neosporin. It may need to be flushed or cleaned daily as needed as well - just depends on what it looks like.
Do not shoot the hawk. Bird of prey are federally protected and they can't be harassed. Look up your laws.