Pull that melted mess off the plug and replace it with another one that's not damaged.
I think that would fix it, sounds like the relay is what controls the DRLs, which use the
high beam bulb during the day, the relay probably controls night time lights low beam.
Unless you are legally required to have DRLs, you could rewire that mess to work only
with the switch inside the trucklet, otherwise find a replacement part
Wild