I drive a 2000 Audi TT Quattro.

I had my oil changed at Firestone on May 8th. On May 14th, as I was driving, my check engine light came on. I pulled in to a gas station and popped the hood, and saw that the air filter cover was not replaced properly. There was about a two inch gap between the lid and the base on one side, and the other side was not properly screwed in. I took it to Firestone that afternoon, and they cleared the code, saying it had something to do with the air/fuel ratio. The next day, however, May 15th, the check engine light came on again. Yesterday and today, the engine has had a very hard time starting up, it has ignition, but it has difficulty maintaining ignition. Could this be caused by driving with the air filter cover open for a week? Could dirty air have come in through the intake and potentially caused damage to my internals?

I plan on going back to Firestone after work this afternoon, and would like to be armed with knowledge regarding the effects of their carelessness, if at all possible.

Thanks in advance.