Unless you can post a schematic for your game we'll have to use one for Deluxe Pinch Hitter from IPDB.org which might be similar:
Deluxe Pinch Hitter Hi Score Impulse relay (resized).jpg
I think the Hi Score Impulse relay (bottom right in red) should fire on this game to advance the high score score reels. For that to happen a path needs to be completed through one switch on the Run Unit and through the three step switches in red.
At the top of the schematic the Hi Score 10 point score reel disc has four positions. Position 1 does nothing. Position 2 (20+ runs) selects a set of 10 contacts on the Run Unit disc. Position 3 (30+ runs) selects the next 10 contacts, etc.
The Run Unit disc has 10 pairs contacts wired together. So for example the contact for 21 runs is wired to the contact for 31 runs, and it's the Hi Score 10 point score reel above that selects between the two.
Below that the Hi Score 1 run score reel selects one of the 10 pairs of contacts from the Run Unit disc.
So for example if the High Score reels show 20 runs and the player has scored 20 runs the path in red should be complete. When a run is scored the Run Unit switch should close which should fire the High Score Impulse relay which should advance the High Score 1 runs score reel.
If you're lucky the Run Unit switch needs to be cleaned or adjusted. If not you'll need to trace through the three steppers.