Almost all high level Artistic gymnasts suffer chronic wrist pain. It’s even ended competitive careers.
If you, like me, think that signing a contract promising to avoid injury won’t work, then consider other options.
Pommel horse is definitely the biggest culprit.
Brain-storming at a recent coaching course, we came up with a few suggestions:
take fewer turns of higher quality
every approach is “precious”
the coach should watch every turn
specific goal for each turn
3 attempts maximum for any specific sequence, then switch to something else
when skilled enough, spend more time on the pommels, less on mushroom
do not do more than 20 circles in a set
The best suggestion of all — and one we did often at Taiso over the years — is to split training into two blocks during the same training session. For example, do skills, sequences and routines near the beginning of training for 15-20min. Then basics, specific drills, single pommel work, etc. near the end of the training.
One new suggestion for me was to train a 30-40min handstand at the end of workout to “toughen-up” the wrists.
Sounds crazy. Firstly, I don’t have 30min at the end of work-out. It’s not specific to my needs in competition. And that drill is more likely to hurt the wrists than condition them.
Most coaches concurred that wrist supports should be avoided unless absolutely needed. Then worn every training session, if needed.

Colored Tiger Paws Per Pair
I see more Tiger Paws than any other brand of wrist support. (With girls I sometimes think they are more of a fashion accessory than essential equipment.)
At Taiso, if the wrists got too, too sore — we would give the gymnast a complete week (or two) off pommel horse. This definitely worked to relieve wrist pain. Unfortunately the pain tended to return once we started training again.