Problem: Accelerator delays

First post! I have a 2010 Elantra with 55k miles and bought it new. For the past few months my car has been acting strange. For the first 20 to 30 minutes while driving, my accelerator would delay and the car wouldn't be in normal motion until after a few seconds of pressing on the gas pedal. For some reason, this has been eating up my gas more than usual. I took it to hyundai service and they said this specific problem is not part of their warranty so they require a big payment to get this fixed. Help!

-FYI: I havent done any maintenance on this car yet. I know this is really bad! but help me out here because I gotta know exactly whats going on with my car.

Thanks