Will an antibiotic for an uti also clear up an infected wound?

It depends.: It is never a good idea to "recycle" antibiotics or any other prescription medicine. Some antibiotics might work for both infections but it is unlikely you would have enough to treat the wound. Also, some wound infections need to be cleaned, drained, or otherwise treated to remove the source of infection.

Sometimes: Not all bacteria are sensitive to the same antibiotics. Sometimes several different bacteria will be treated successfully. The presense of one infection could be a cause of another in which case the likelihood of treating both with the same agent is higher. Be sure you take it as directed and take all of it.