Yup, that looks like A-4862 coil stop which is stocked at Pinball Resource. The Player Unit also uses it.

That's to keep the transformer quiet. Sometimes the windings inside the transformer become loose and cause humming/buzzing noises. This method of wedging in plastic or wooden wedges silences the noises. Leave them alone.

We use those woodworking syringes to put wood glue between the lifted layers. Then clamp it down and let it set for 24 hours.