I would defently start by fixing the connector...
Because it is broken could be entering water or making bad contact/contacts deformed or something as obviously it suffered some "event".
I dont know, surprised the garage ignored that.
The connector on the left (T4b)has 4 pins, two of the wires connect to the CAN system, one pin is not used, the other comes from fuse in dash. This switches pump on. These wires are quite delicate. Don't allow the pump to hang by these small wires.
The centre connector (T2)is the + & - power from the battery. the 3 pin on the right (T3f) is the steering angle sensor. (some cars don't use this, and some cars have a 4 pin angle sensor)
It seems likely that the first time the pump failed, which can happen, whoever removed it first, damaged the connector T4b, because it is delicate. Probably all the subsequent failures originated from that. If VAG dealer has fitted several pumps, surely they would have seen the condition of the connector. Basically , if the first person damaged it, then its negligent, and if successive people saw it but did nothing about it then they are negligent.
There is a knack to undoing the connectors. The damage kooks pretty bad, and if the latch is not working, then the connector probably will vibrate loose, hence intermittent operation even if the pump is ok, and not only will the connection be bad, but water enter the pump side of the system. Or there could be a loose connection within the connection wire in the connector, or even a broken pin because of the damage.
There is a fuse under the RHSide dash which switches the pump on. possibly fuse number 1, a 7.5 amp entitled "power steering/engine operation"... What that means is... engine is operating so send signal to the powersteering pump. You should have voltage at that fuse when the engine is running, and none when engine stopped. check fuse, and possibly put multimeter ont + side of fuse to see if voltage appears when the engine starts running.