To prevent wrist pain during push-ups try one of the following simple tips:
Modify Your Hand Placement
Experiment with different hand positions. For example, you could try placing your hands slightly wider than shoulder-width to reduce wrist strain.
Use Push-Up Handles
These low-cost items allow you to maintain a neutral wrist position (rather than flexed) while performing push ups which usually eliminates (or greatly minimises) wrist discomfort.
Easier Progression
Start with easier variations like knee push-ups which put less force through your wrists and gradually work up to standard push-ups as your wrist strength improves.
By following one (or more) of these tips you should be able to minimise wrist pain and get maximum benefit from your push-ups!
If none of the above help it might be worth consulting a fitness professional or physio to dig a little bit deeper into why you are experiencing wrist pain during push ups!