I wonder if this red switch on the O relay is staying closed when it should open. If I understand this circuit correctly, the Ball Return switch should energize the Q relay. A switch on the Q relay (which is now closed) in series with the red normally-closed O switch and the Motor 1C switch (the red circuit) would turn on the P relay. The P relay turns on the score motor for a cycle, which increases the ball count by 1. A switch on the P relay also energizes the O relay when the Motor 4C switch closes - opening the n/c O switch in the red part of the circuit. With that O switch open, the P relay should de-energize when a Motor 2B switch in its lock-in circuit (in green) opens (at the end of the motor cycle)...unless that red O switch is staying closed, in which case the P relay would stay energized and keep the score motor turning, and the ball count increasing.

When you remove the ball, you open the Ball Return switch, which turns off the Q relay, which then shuts down the P relay, which shuts down the score motor.

Is the P relay staying energized the entire time this ball count increase is happening? Is the O relay energizing at all?
PlayBall circuit 3 (resized).JPG

Yeah, that O coil looks pretty dark. I'd unsolder one of the 2 leads and measure the resistance with an ohm meter. I'm guessing your coil will read open. You can get those coils from The Pinball Resource - that coil is an A-9735.

Once you get it in there, you'll have to watch it to see if it locks on - it burned up for a reason.