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2004 Hyundai Santa Fe - Veers to the rigth

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.

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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.

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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.

Tester