Santa Fe makes jerking motions

Sometimes a loose gas cap will activate the chk eng light. Forget it.

Check the transmission fluid level first and make sure it's full. If it is full then a trip to a transmission shop would be the next step. It's quite possible that a simple fluid and filter change may fix the problem.

The first thing you do is to have the fault codes checked with a scan tool.

it has to do with the injection system. Might be something stupid like a manifold intake gasket leak (commonly known as cylinder 4 misfire problem) or it could be a part of the system like : spark plugs, wiring, fuel injector, throttle position sensor. A low cost solution worth trying is a kleen flow injection system cleaner. It will remove a lot of carbon and see if it helps...

Make and model, year, engine, transmission? 2wd/4wd, ever been wrecked, etc.
could be drivetrain parts such as c.v. or U joints.
could be fuel system clog somewhere.
could be a bad sensor.
We will need more detailed information.

Sounds like a transmission issue. Have the transmission checked. I assume this is an automatic.

Are the engine rpms climbing when its jerking forward? seems like it could be a Transmission torque converter problem.

Scan for codes. Discribe when it jerks! Under a load,going up hill,pulling away from a stop? Does it shift okey? Possibilities--Bad ignition coils, Bad Throttle Position Sensor, Bad EGR Valve,Dirty Throttle Body ETC.

sounds like you need a transmission, with those incorrect gear ratio codes

i have a 2004 hyundai santa fe and i've had the same problems. it's so random when it happens but it seems to be when its a bit warmer outside.. above 70 degrees. it's when i press the gas from a stopped position the car just almost dies. i've pulled out in traffic and thought i was going to get killed. then the car accelerates and the tires squeal and the engine revs super high. the car jerks and its very scary. i bought the car new in 2004 and it started doing it after a few years and it still does it. it's not the gas pedal. it seems like a glitch in the computer or something. it's very dangerous!