Can I have a plastic surgery procedure or anything else to take away the look of my prominent veins on the back of my hands?

Many options. There are a variety of options ranging from ablation techniques (with sclerotherapy, laser or stripping) to camouflage with injectable fillers or autologous fat injections.
Yes . I agree with with dr fiala that a filler on the back of the hand will help conceal the veins there. Another and often preferred option is sclerotherapy to veins on the hands. The sclerotherapy removes the veins permanently but usually takes a few treatments. The fillers including fat , Perlane (dermal fillers) or juvederm also improve the appearance of thining skin.
Fillers. On the back of the hand and ipl are also used to back the hands look more youthful.
Advise Against. Although you can always find a plastic surgeon who could remove your hand veins with sclerotherapy or micro-phlebectomy, as a vascular specialist I advise against. These are NORMAL veins & serve purpose for venous drainage of hand, allow for blood draws & IV access. Better option is injectable fillers or autologous fat injections, although these usually provide less & more temporary improvement.
Removal of prominent. It depends if you have very large veins on the back of the hands and minimal soft tissue. I remove these veins through tiny puncture skin incisions. I also fat graft the back of the hands to give a more youthful appearance. Many times, you can tell the age of a person by just looking at their hands.
Eradicate hand veins. Prominent hand veins on the top of the hand are a common problem . I prefer sclerotherapy for the smaller veins and , if the vein is large enough to accept a laser fiber, then i perform laser ablation of the vein just as we would do for the saphenous vein in the leg. This procedure has minimum discomfort, minimum bruising, no downtime and excellent cosmetic results.
WHY. Why in the world would you want to do anything? Ask a trusted friend if it is noticed by any body.
Veins. Usually you don't have the veins removed but if you are lacking soft tissue on the back of the hand you can either have a temporary filler or a more permanent correction by fat grafting to the area.