See a doctor! Causes of wrist pain are numerous and you while you mention you have tendonitis, I don’t know if that is self-diagnosed or not. Have a professional look at it.
In the past, my clients have had success with chiropractors who are skilled in periphery limb adjustments. Athletes certainly saw great benefits too and I did as well with my feet and ankles.
Most common strengthening exercises learned, in the Pilates-arena anyhow, are in supination and pronation of the forearm. However, working the wrists in all planes and axis with slight resistance and also stretching is often needed. This may be your case.
Google "wrist exercises" and you will find a plethora of information. I found the University of Michigan and the Mayo Clinic sites quite informative.
As far as executing Pilates exercises, watch for: equal weight distribution on the hands, lifting up and out through the center of the hands and up through bones of the arms, caution with loaded wrist extension on the harder exercises like Snake and Twist. These latter exercises may cause a lot of discomfort and pain if done incorrectly. One must lift the rib cage out of pelvis on weight bearing exercises to help unload the shoulders and arms always.
Good luck and do let us know how things work out.