Could doctors give an anesthetic while stitching a small wound?

Most often a: Local anesthetic is used for this situation.

A local anesthetic: is usually given just prior to suturing most wounds unless a single suture only is required. The discomfort from one suture is less than injection of a local anesthetic

Most hospitals: Including emergency rooms are set up so the nurse can give you sedation while the doctor sews you up with local anesthesia. The local can be given though with a very fine needle but does burn a little when going in.

Yes: A local anesthetic can be used to numb the area before stitches are placed.