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I would put the relay back in, as I do not know what relay that would be, the neutral start relay should be on the relay panel under the dash in position #4. Unless you can identify where the noise is coming from, I would ignore it for now.
First off you must check at the starter #50 terminal (starter control) for battery voltage when the ignition switch is held in the start position. This is the small wire pushed on the spade connector. The only other wire is the positive battery cable. If you have at a minimum 11.5 volts, the starter has failed and requires replacement.
If you have voltage, but less than 11.5v, get back in touch with me and we can proceed down that path.
If you have no voltage signal and it is an automatic, I suggest you try jiggling the shift lever in and around the park and the neutral positions with the ignition switch held in the start position. If the starter cranks, you have a faulty or misadjusted neutral safety switch. Remove the under dash panel at the driver's knees. Turn the key to the start position and hold it, move the shift lever in and out of park you should hear the relay click (position #4) click. If it does click, cantact me and we will proceed down this path.
If you still have no reaction based on this year model, it has been my experience that the electrical portion of the ignition switch has failed. This IS the most likely cause of the condition. Another "hint" that it has failed would be a "notchy" feel when turning the key or poor return spring tension from the crank to on position.
The switch is just the electrical portion, not the tumbler your key goes into, or the steering lock assembly, but the electrical portion mounted on the back inside the dash. It is accessed by removing the instrument cluster. The part is about $35 US and about one and a half hours labor to replace.
This year model has a lot of ignition switch problems and Audi has come up with an updated switch. The part number of the switch is # XXXXX 905 849 B. I suggest you purchase one, remove the instrument cluster. You will see the switch in the dash directly behind the key lock assembly. If you first disconnect the wiring connector and plug it into the new switch, you can test it's operation with a screw driver in the end of the new switch. To replace the switch you will see two small set screws under some dabs of red paint. With a small straight screw driver you can loosen these screws and install the new switch. Be sure you turn the new ignition switch back to the off position so the slot position matches the switch you removed.
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Jake "THE AUDI DOCTOR"