Can hand sclerotherapy create new veins or make existing ones more prominent/larger?

Unless you have a problem with the deep veins of the hands (which is unlikely), sclerotherapy on the dorsum (back side of the hand) should not cause you to form more veins on the wrist. I suggest that you see the treating physician for a proper evaluation. Watch the Vein Treatment Center video about varicose veins of the hands on the link below.

Sclerotherapy works by the sclerotherapy solution pushing the blood out of the vein and the solution irritates the walls of the vein so that they stick together to dissolve the vein. A side effect of sclerotherapy could be new vein formation (neovascularization). This occurs occasionally when veins are closed and new veins grow. This seems to be what has happened with you. You should follow up with your treating physician. I usually treat the larger hand veins with a laser and and the smaller ones with sclerotherapy. This combination avoids the new vein formation most of the time. If nay new veins form then they can be retreated by either option depending on their size. You may need a second treatment.