Not an easy problem to solve, Steve. Below is a pic of an undressed Gicar for the Anita - Andreja should be VERY similar, if not identical.

The transformer takes 120 VAC and has taps for 18 VAC and, I believe, 5 VAC. The 18 VAC is rectified to DC to operate the double pole single throw (DPST) relay coil and the 5 VAC is used in the autofill sensing circuitry. One of the poles of the relay supplies power to the fill solenoid and the other powers the pump - obviously simultaneously during autofill operations.
The Gicar controller in the Quickmill line has been very reliable - not that it can't fail - but that this would be the first instance I can remember on a QM machine - corrections strongly invited. With that in mind, I would first look at some other "things" because the sound we all hear when autofill takes place is that of the pump. Are the electrical connections to the pump secure, is the little "Klixon" thermal diode on the pump secure in its pocket?