Can doctors at a medical clinic sew stitches from a cut to the wrist or does this need to be performed at a hospital?

Stitches: Yes, most doctors can

Depends on the: clinic setup and the skills of the attending physician. Though will think serious injuries should be attained to in hospital ER

Office v Urgent Care: While I can't speak for Australia, many smaller private practices in States still do quite a number of things/procedures in the office, such as sewing up wounds. In larger groups, due to time, space & staffing issues, a specific urgent care clinic is set aside for these types of walk-in appointments. Depending upon site of injury & severity, some pts are sent to ER for care under Plastic Surgeon.

Clinic can stitch: You don't have to go to an emergency department at a hospital for stitches. There are now urgent care centers in most parts of the world that offer non life threatening, acute care for a fraction of the cost of a hospital. A clinic that requires appointments is a bad idea, though, because lacerations can only be closed within 6 hours.

Possibly : Some clinics and urgent care centers will do simple sutures. Depends on the location, bleeding, associated anatomy ( lots of important stuff in that location, blood vessels, nerves, etc). We could determine extent in a prime visit and point you in the night direction, certainly. Best of luck!