I would first advise that if doing push-ups hurts this much stop doing them, for now. A question would be, did you notice when this first occurred and what you were doing? You could have an injury that just needs some time to heal. It's hard to see why having wrists that aren't that flexible would lead to pain. Flexibility allows us to do things better, but it doesn't necessarily have much to do with pain unless you did something that turned a wrist farther than it could go and strained it. The only way to know for sure exactly what it is, assuming that's possible, is to see an orthopedic surgeon who will take X-rays and perhaps an MRI and also manipulate the wrist to see what movement makes it hurt, but that doesn't always lead to a diagnosis or a correct diagnosis as the body is complicated. Usually then you'd be sent to physical or occupational therapy to work on it, which is where you go to learn from people with actual training what exercises might help. But without a diagnosis, that's hard to do, as you wouldn't know what is causing the wrist to hurt in the first place. The easiest thing to try is to stop doing those push-ups, and rest, elevate, and ice the wrist and see if in a couple of weeks it isn't better.