The switch with the missing white-blue wire is the End Of Stroke (EOS) switch for the Score Reel. When a Score Reel should advance its point relay (10 pt relay in this case) fires and closes a switch that sends power to the Score Reel solenoid. The point relay stays closed until the EOS switch on the corresponding Score Reel opens which cuts power to the point relay, and in turn, the Score Reel solenoid.
The EOS switch is there to make sure that the power to the Score Reel solenoid doesn't cut until the Score Reel plunger has pulled all the way in. It's a feedback mechanism to make sure that just the right amount of power is sent to the Score Reel and no more.
It sounds like your Score Reel plunger isn't able to pull all the way in for some unknown mechanical reason so the EOS switch never opens and the point relay never stops delivering power. Rather than fix the problem someone disconnected the EOS switch so that you only get a brief pulse of energy to the Score Reel while the 10 point target is closed. It can work that way but it's unreliable.
Once you sort out the mechanical issue with the Score Reel reattach the EOS switch and you should be good to go.
/Mark