First thing first. Can you hear the air injection pump running? You can best test this out in the morning during a cold start. The air injection pump should run for at least 30 seconds, if not longer; and needs to eventually stop. You mention there is a pending lean fuel trouble code. This may or may not be due to the air injection pump relay being stuck on. So... listening to the pump is the first step. If it's constantly on, then you'll need to diagnose why the pump is constantly. Is the ECM (engine control module) commanding it constantly on, or is the relay stuck. The air injection monitor will not run if any trouble exists within the air injection system, or if the ECM senses a lean fuel condition. The air injection monitor is tested via commanding the air injection pump on, and checking the oxygen sensor values for lean condition. Constant lean, no monitor set. No oxygen value change, no monitor set. Your first mission will have to be finding out if the AIR system is working as it should. Once it's ascertained that the pump does work as required, driving your Audi tt through a drive cycle should complete (make ready) the air injection monitor and others, and allow your Audi to pass the smog test.