upon acceleration goes fine with pull to right and when gas is leveled off it will veer to the right and back.
Front wheel drive vehicles exhibit a phenomenon called “torque steer,” where they pull to one side under acceleration. I presume you’re not talking about that. Bad alignment can cause what you’re describing. So can a dragging brake on one wheel. Or a low tire. But all of those should be at work all the time. Take your Santa Fe to a good independent shop that specializes in alignments and see what they say after they look it over.
Sounds like your lower control arm bushings are torn/loose. Loose lower control arms will shift when accelerating and braking and cause a change in the steering.
I’ve also seen torque steer caused by a dry ball joint due to a torn cover boot.
Check for worn engine/transmission mounts.
I just changed a worn one in my Honda for torque steer when taking off from a stop.