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If this has been going on for a month, I would have your vet evaluate the area.
It may be a tumor, or may be a multi-drug resistant infection
A culture and biopsy can be considered.
It may need to be surgically debrided to remove the dead tissue as well
would antibiotics be helpful if it is an infection?
Yes - it would
Is there anything else I can do to help the wound?
This large - unlikely.
Best to get to your vet, especially since this is this long now...
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