I was able to see the pictures. The burnt connector explains everything. That is good and this fix will be quick. All you need to do is to cut the damaged connector as close to the connector as you can. Then, peel some of the plastic wire protector in order to expose two to three inches of the three wires. Remove about half-inch of insulation at the tip of each wire. If any copper wire strands look burnt, cut some more until you have clean copper strands in each wire.
From the new spare connector, remove the mud from the connector and cut the wires about six inches from the connector. You will also be using some of the plastic protector flex tubing to cover these six inches later on. Remove half-inch of insulation in each wire. The copper of each wire should be clean free of any green/white/brown oxidation.
Get some heat wrap tubing or sleeves at your local Autozone. Select the closest diameter to the wire insulation. The sleeve has to fit loose and not tight to the insulation. Use about 2 inches of heat wrap sleeve for each wire. Insert the sleeves at each wire in the new connector. Place the sleeves at the connector side away from the tip of wire to be spliced. You can also use a bigger diameter heat wrap sleeve to cover all three wires around if you want.
Now, match color and position of one wire, hardness and connector side. The wires should match in color and strips, red with red, black with black, and yellow with yellow. Please them one in front of the other with about 1/4 inch of copper passing the insulation of the wire in front, twist the wires together until they are secured, and solder the union. Do not apply too much tin. The trick is to prevent sharp or pointing tin spots in the union to prevent pinching the insulating sleeve or heat wrap. Once the wire is soldered, place the heat wrap sleeve in the union so the union is in the middle section of the sleeve. Now apply heat to the sleeve using a heat gun or rubbing the tip of the soldering iron in rapid back and forth movement until the sleeve is tight in the wire. Repeat this process with the other two wires. If you have a larger sleeve, then use it to cover all three unions. This sleeve would be 3 inches long. Finally, protect all exposed wires with the black plastic flex tubing and seal with electrical tape.
I hope I explained myself correctly in the procedure above.
Finally, you need to find out why the connector was damaged. Make sure the headlight bulb is the correct power. A larger power bulb will overheat the cable and connector. Make sure there are no short circuits or other wires when connecting the headlight again.
Let me know how it goes. I will post tester information later.