“What can be the cause of my wrist pain?”

Without injury, usually tendinosis. Rest, rehab, bracing will help.

Something you’re doing is straining your wrist joint. Make sure when you are doing chest or shoulder presses that you keep your wrist straight and not bent backward. Give it some rest too and train lower body. If it doesn’t get better see and Orthopedic hand specialist or your family doc

There are multiple possible causes. Stop doing the exercises in the gym that are causing your wrist pain and focus on other exercises such as a bicycle or treadmill for strengthening of the lower extremities. If the pain does not rapidly reduce, you should consult a trainer, physical therapist, family physician or sports medicine specialist.

The most common reason is what would be called dorsal impingement. This is simply irritating the joint capsule because as you lift you extend your wrist and pinch this tissue. Also, could have an early stress fracture involving one of the bones of the wrist. It would be unlikely that you have significantly sprained one of the ligaments, but certainly these could be stressed causing inflammation. These are the most common reasons. I would try wrist splints or tape thing or weight lifting glove to limit extension of the rest. I would ice the rest after working out and also stretch prior to working out. If you continue to have problems with it, have an X-ray taken and an evaluation by an orthopedic surgeon.
Good luck.