Strange starting issue

Hey guys, long time no post! I've recently had an issue with my A6 2.8 FSI - long story short, the car would start but the moment I switched it off (irrelevant if the engine was cold or warm) it wouldn't start again straight away unless I waited around 20mins, sometimes I had to wait overnight. On the 3rd day it completely died and wouldn't start at all, so I took it to an Audi specialist who spent 2 hours to find the fault: a wire behind the dashboard that caused short circuit, they fixed it ("apparently") and the car started again. After driving the car for 9 miles, I stopped and guess what, it wouldn't start - I could hear the fuel pump prime and when trying to start I'd hear the click noise from the LPFP, but no crank at all. I waited 20min and the car fired up, drove straight back to Audi specialist but couldn't replicate the issue as the car would start on all 20 test attempts. They read some codes but were like: oh yeah we'll need to spend at least a few hours trying to find the fault as it can be very hard bla-bla-bla; I felt like they're trying to milk me so I went home.

A few days ago I started the car and drove for 10 minutes, stopped, tried to start again and nothing. Waited 20min and it fired up without any issues and even if I drove again and switched it off, it would start without any issues. No warning lights.

Yesterday I started the car and immediately after starting it began to shake vigorously for like 8 seconds (hunting revs). Someone managed to help me read the codes with their VCDS and this is what came up:

Starter control, feedback terminal 50
Short circuit to ground/interruption
Code: P3053

Engine
Starter relay electrical fault in the circuit
Code: P3088

My understanding is that this is relay related? Any help would be much appreciated.

Ermmm, anyone?