Low beam plug

Just a possibility

Maybe the actual connector(s) inside the plastic are Sprung (not gripping its mate tightly) or corroded. This "loose" connection would get really hot and melt the plastic housing.

An auto parts store should have that whole pigtail and also have butt splices to connect the wires back together.

Also check the light housing for it holding water (may be cracked) water sloshing on the hot light.

I am banking on my first guess!

Let me know what you find!

The Medic