My rear turn signal socket connector has a broken wire as you can see in the picture. What is the best way to repair this issue. Can the wire that is broken be pulled from the connector and then reinsert the remaining wire from the harness? Any help is really appreciated.

 

Source a replacement connector, new or used. If new, cut and splice the entire connector. If used, cut and splice the entire connector, or remove that terminal and cut and splice just that wire. What vehicle is it? Because that is not a taillight connector for an XJ. Therefore, i do not know how to remove that terminal in yours.