First, make sure the score reel is working properly by manually rotating the score reel. Push in the solenoid plunger and see if the score reel clicks into place as it rotates to the different number positions. It should smoothly rotate as you push the solenoid plunger.
It looks like the solenoid overheated at some point. This doesn't mean the solenoid is bad. You can check the solenoid with a multimeter on the ohms setting. The coil should have 2.0 ohms or above. If it's below that, then the coil is weak and it should be replaced.