Golfing injury of left hand, pain for 4 weeks in region of ulnar nerve...could it be fractured hamate?

Impossible to say: How can you expect anyone who is unable to examine your wrist and order appropriate imaging studies to tell you for certain? They are many more likely causes of your pain than a hamate fracture. See an orthopedist and have the wrist evaluated.

Wrist pain: It's hard to know, but if you are right handed, a tendon or ligament injury is the most likely. An exam and an xray would help. Good luck and get better!