Hyundai - Accent :: 2007 - Occasionally Jerks Upon Accelerating

My 2007 Hyundai Accent Automatic has been, what I'd call jerking, off and on when I go to accelerate, either when speeding up to merge on to the interstate/coming out of a turn/or being very slow to accelerate after a stop, like at a red light. About a month ago, it wouldn't start after I filled it up with gas and the check engine light finally came on, so I took it a mechanic. They kept it for a day, got it duplicate the not starting problem, and said it was the "fuel gauge." (looking this up on the internet leaves me questing, because nothing about the definition of a fuel gauge sounds like anything that would cause it to do what it was doing, but they told me that's what the computer alerted as the code for what was wrong.)

It was working fine now when I fill it up with gas, but occasionally still jerks upon accelerating. I haven't looked at the gear gauge to see if it's not changing gears, as I thought at first it might be something misfiring, but I had the spark plugs changed probably a year ago or less. Just based on a quick internet search, I think it might have something to do with the fuel pump. I'd describe the jerking as just that, the car physically jerks until either the gear changes, or it gets power or something, and then once it does it it doesn't accelerate quickly after being at a complete stop, and the engine seems to get louder and while it does move, it seems like when I push the gas petal it has a hard time getting going.