Can a cut to the wrist recover and be well enough to play again?

Yes: Of course it depends on the depth and the extent to which other structures such as tendons are injured. In general, as long as an infection does not occur, a patient can expect full recovery from a simple laceration.

Need more info: Depends on what is cut (skin, nerves, tendons), what treatment you have had, and what you want to play. If it is only the skin, you will probably be able eventually to return to any kind of playing you want.