MAF = Mass Air Flow
AMM = Air Mass Meter
Basically the Air Mass Meter measures Mass Air Flow. People exchange the terms, and they generally mean the Air Mass Meter. This is the sensor straight after your air-box.
Hesitation is a common problem on these cars, in fact any 1.8T engine.
A feeling of lack of performance is also common.
My car has had a hesitation problem in varying degrees since new. Just cannot get to the bottom of it. But in summary it could be caused by:-
Vaccum/boost leak somewhere
Faulty OEM Diverter Valve - these tend to stick, or the diaphragm splits
AMM/MAF - they degrade, and almost never give a fault code/engine warning light. Only when they really do break, do they give the engine warning light.
Coolant Temp Sensor - make sure you have the latest one
N75 - this controls boost basically, by contolling the vaccum which in turn controls the wastegate actuator.
N249 - this controls the vacuum going to the top of your Diverter Valve. So that the ECU can tell the Diverter valve when to open and close.
Coil Packs - You can have an old version, and one could be on its way out. Check the versions. Or just blanket replace them.
Spark Plugs - S3/TT spark plugs are supposed to last 40K miles. They do. But some people like to replace them more regular. One of the simplest things to check is that they are torqued up correctly. If they aren't, they could come loose under continual cooling and heating. I've seen a post somewhere, where plugs were not tightened correctly and the guys coil packs looked like they'd been under a blow-torch.
02 Sensors - Upstreem and Downstreem of CAT
others:-
Throttle Pedal Position sensor
Brake pedal sensors - 2 of them
Clutch pedal sensor
Throttle body - very rarely replaced. Sometimes it just needs a clean inside with some carb-cleaner, then perform a throttle body reset (TBA) with VAG-COM.
There are a lot of sensors which feed into the Electronic Power Control system (EPC) and the Electronic Stability System (ESP) which if faulty, could cut power slightly. E.g. the throttle, clutch, brake sensors => EPC, Longitudinal, Lateral, Steering angle => (ESP).
Its a mine field. I've given up now to be honest. But some of the simple stuff like plugs, coil-packs, air-filter, AMM, DV, can all be replaced quite easily, and should result in the car running better.
Good luck.
AL