Mac3rd2012

(Sorry if this ends up being something obvious, I've been looking everywhere online trying to find a solid answer)
Ok so I've been having this incredibly frustrating issue with my Stang. It's a 96 GT, and lately it's been just not starting. Doesn't crank, no clicks, nothing. Battery and alternator are brand new and have been tested and are good. Starter relay also has been replaced. When I try to start it, and turn the key to the on position, the dash lights up like normal, then when key is turned I get nothing. I'll leave the car for a few days, go back to it, turn the key and it fired right up. I've taken every ground I could find and cleaned the connections. The only thing I've found that replicates the problem is there's a small black clip/plug that goes on the starter solenoid. There's a little prong that comes off the solenoid, almost like a spade connector. When I unplug it, the car will do what it was before, no crank, clicks nothing. I plug it in, fires up. Also when the car wouldn't start I checked the wires on the starter and nothing was loose. I randomly unplugged that wire and plugged it in and found it made it start. So my question is, what exactly is that black connector/wire for? And could I just cut the plastic part off and crimp a spade connector on it? As I've found that they do fit on that little prong. I've tried tracing it but these cars aren't the most spacious when trying to backtrack. I'll include a picture of what I'm talking about to hopefully help get an answer

Mac3rd2012

Bueller? Lol anyone