Unless it's a mechanical bind in the relay, I don't think cleaning it will help. The schematic can show how the circuit works.

Green line:
The ball closes a shooter switch. The circuit also goes through motor cam Index D. That turns on the shooter relay. The shooter relay turns on the score motor and other circuits tally the score.

Blue line:
Index D will open as soon as the score motor begins to rotate, and the shooter relay must remain on to complete the sequence. The relay is held on by its own shooter relay switch through cam 5-B.

Violet line:
The shooter coils are pulsed via cam 4-C, through the shooter relay and cam 5-B. Since the ball is no longer on the shooter switch, it opens.

A moment later cam 5-B opens to drop off the shooter relay to end the sequence.

ckt.png
Cam5.png

*The most likely suspect is the PF shooter lane switch(s).
* The next likely suspect is that cam 5-B does not reliably open.

Knock on wood, I had this same issue develop last week.

All the switches outlined above checked good and I began looking for odd things like solder tabs touching each other. Then I noticed turning off/on the game occasionally didn't clear the issue: the kickers would begin kicking at power-up.

polyacanthus post was pretty close.
The shooter relay armature was clearly sticking onto the magnet pole. But neither the pole nor the armature was magnetized. It was not a mechanical bind.
It didn't stick by manually pressing the armature, it had to first be energized.
What's more is the brass insert in the relay frame where the coil's mounting screw passes was in good shape. (See the note below)

I didn't detect any kind of sticky substance on either the pole end or the armature. I didn't detect any wear on the parts.
I removed the coil and cleaned the armature and the pole real good with automotive wax and buffed with a clean rag.

Cleaning these two parts with polish fixed the issue, in my case.

Note for general information:
I used to have a *terrible* persistent problem with a sticking match relay.
The arm would pull in and magnetically stick to the pole and had to be pulled off with my finger tip. Upon disassembly and inspection, no magnetism was detected what so ever on either piece. Cleaning and adjustments made no improvement.

You can see the effects of this defective match relay on my demonstration video. The upper lane and Special lights are not switching properly:

Answer:
The coil's steel center pole was in contact with the steel relay frame. That caused a temporary magnetism condition until the armature was forced off the pole by finger tip.

It took a long time for me to discover this, that my relay was missing the thin brass orange washer in the diagram:
Diagram.jpg

I got a brass washer from my junk pile, but the (normal thickness) washer was too thick so I grind it down to size.
Issue fixed.

On the shooter relay, the brass washer is part of the relay frame, pressed into place by the factory.

Great you got the shooter working. If you bump that thread about your super advance then I'll help when I can.