Hard to tell at that angle. Is it a vacuum hose?
Thats the vacuum supply for the anti shudder. When you stop the engine that actuator closes a butterfly flap in the EGR valve that stops air going into the engine. Because no air is going into the engine as it slows down it can't compress it so the engine slows a lot slower and more smoothly.
Without the flap closing the engine stops a lot quicker and can rock the car a bit.
Car will work perfectly well without it, a vacuum is only present for a few seconds when the engine stops. if you have the bonnet open and stop the engine you will see what it does.
When I deleted the EGR on my car it had no valve and it didn't cause an issue.
Karls 312bhp A4 3.0 TDI build thread http://www.audi-sport.net/vb/audi-s4...ld-thread.html