My A8 won't start!! Can't you get gas???

Here's a problem description:

With my A8 4.2, I noticed the last time he jumped a little worse than usual. And now today, my wife was driving the car and after a strong right turn, he was running very slowly. Since it was just a little bit to us home she wanted to see that she still gets the car home. It came as far as the engine went off... she could then start it again... drove another few meters...the engine went off again and now he can't start anymore. It's not the immobilizer. I'm fine with the gas pump. My suspicion now lies either with the relay for the gasoline pump or the gasoline pump itself. Could it be a clogged gas filter?
Meanwhile, the battery is now empty due to the futile start attempts.
Does anyone have any useful tips for me or has anyone ever had this case and could help me further?
Thank you so much!
P.S. I didn't really want to put my Christmas money into an expensive workshop stay...well, yeah :-)
And there's something else in the back: does any of you know what to look for if you want to buy an Audi TT (no matter which engine)?