Fuel Trim issues
There are a few different reasons on why you could be a getting a check engine light due to fuel trim. I think I've seen just about every type of issue, and some of the steps I used to resolve/diagnose.
On the 1.8 the most common issue for a lean condition has resulted from cracked vacuum hoses. The 1.8 engine bay seems to get hotter than the 2.8 due to the turbo, and it takes it's toll on the little rubber hoses. Check the hose that goes from the manifold to the fuel pressure regulator (FPR) and also the one that goes to the bypass valve. Cracked/frayed hoses are a very common problem. The good news though, vacuum line is cheap and easy to replace.
Mass air flow (MAF) sensors are another known culprit for causing fuel trim issues among other problems. Audi used a few different MAFs throughout the years, some were known as a wire type, the other as film element. The film ones seem to be more prone to building dirt on them and failing vs. the wire type. You will first want to check your engine load using vag-com. The block/value may be slightly different from year to year and model, but there should be a block that shows how many grams/sec of air the car is pulling in. Vag-com shows what the car's normal readings should be under that box. If the values are off, your problem could be the MAF. You have a few choices. You could either just replace it with a new one, or you could trying cleaning it. Cleaning is relatively safe, and often solves the problem. Cleaning the sensor type that uses a film element proves to be a challenge sometimes, and often replacement is the only way to go. There are a few different methods for trying to clean these. You'll need a tamper proof torx bit to remove the sensor from the housing. You can try isopropyl alcohol in a ziplock bag and swish it around submerging the sensor. My personal favorite and seems to be most effective is to pick up some electronics cleaner and spray that in the sensor housing where the air enters. If you have the wire type, this is very easy to do. After trying that and letting it dry out, you can test it back on the car. If it doesn't make it better, try cleaning again. If that doesn't work, it may be time to just replace it.
Coolant temp sensor is another issue that can cause this. Although usually when these go bad, it causes the engine to get incorrect fueling at cold start and makes the car not start. This is a fairly common part to fail, but at least they are inexpensive and easy to change.
Bad fuel pressure regulators is another cause. Although not as common, and sometimes hard to diagnose, these too can fail. If you have checked all your vacuum lines, and cleaned/tested/replaced the MAF, this would be the next avenue I would pursue. Most people don't have access to a fuel pressure gauge, and the cost of FPRs are low enough it would just warrant a quick and easy replacement.
Week/dying oxygen sensors can also be a cause. If your car has over 60K, they are probably due anyway for a replacement. O2 sensors are not intended to last forever.
Bad fuel injectors is another cause. This is even less common then the above and I've only seen this on cars that have been modified. I've had some terrible luck with RC Injectors on 2.8s in Texas. I say in Texas, because I believe we have a slightly higher sulfur/additive content than some other states have, and could cause the issues I've seen. RC tested multiple sets of bad injectors and could not find nothing wrong with them. But when reinstalled in the car, the fuel trim was too rich. Yet, when I installed a brand new set of RC injectors (and Bosch injectors) the car ran fine. I would never recommend RC Injectors to anyone based on this experience. I've never had this issue with Bosch injectors. Unfortunately, this problem is terribly hard to diagnose. You can't easily test the injectors yourself, and when RC tests them they tell you they are fine. Basically you have to rule out all the other variables before you go down this path.
Wiring harness issues is the last I've seen and the most rare of them all. I've seen one 2.8 that had the MAF wires inside of the harness breaking up. This was a difficult one to diagnosis as well, keep in mind the harness looked fine. I had the car running fine for a couple days, then out of the blue, fuel trim values went really rich. I started yanking on things and poking around the engine bay, and when I grabbed the harness that leads to the MAF the values suddenly got better. I cut open the protective sheath, then trimmed back some of the covering on the wires them selves. I soldered new wire in parallel with the existing wires and taped it back up. All was well after I did that.
After you've fixed your fuel trim issues, it might be worthy to check and clean your spark plugs. They could get fouled out if your fuel trim was too rich.
Last thing, if you want detailed info on fuel trim, and engine diagnostics, I suggest reading through the below document. You will have a very good understanding of the factory system after reviewing it.
Advanced Engine Performance Diagnostics
Picture of OEM hoses and parts.