It’s a classic design fault which seems to effect the vag 130 bhp engine more than other tdi's. Some cars are ok but others shudder esp during warm up 70-90 deg c. .
I ve done a lot of internet research for the exact cause but no one (inc Vag group) know for sure. Some blame the Dual Mass flywheel but ive read threads that the DMF is replaced with a solid wheel but the shudder still remained.
Others think it is caused by under fuelling at idle,(I agree with this) as the engine leans out the mixture to increase mpg. There is no easy adjustment to cure this. Seat servicing can raise the idle by a small amount (50 rpm) but that will not cure it. I have heard a diesel specialist can individually tune the injectors to smooth out the idle but that’s going to cost. Iam not sure if a remap will cure it either.
Also the Mk4 has rather weak engine mounts by design and this does not
help when the engine rocks back and forth.
I have found that turning on the air con and rear screen together helps quite a bit as this puts an extra load on the engine and the computer increases the fuel ratio.
If you search on here for “rough idle” or “shudder” you will find out more.
If you Google it you will find a lot of talk in USA on the VW/ Audi forums and some have tried to reject their cars because of it.
Best of luck