It looks as though this is a 5 pin connector, and you are saying that the second pin from the top is not connected but the black wire looks like it’s in that spot. Black is often used for ground, is it possible that the second pin connects to ground instead of one of the through holes? It also looks like there are labels right below the holes that tell you what they are for. If I had to guess, black is ground and red is VCC. The other two are a bit of a mystery, but if you know that the blue wire goes up top then the green is probably the LED. This is all a guess, I don’t actually know. So from the top it might be blue, black, green and then red. You could theoretically just clip the crimps off the wires and solder them in, it’s the quickest and cheapest way. The more correctly way to fix it would be to identify the connector, and buy a crimping kit for that style connector. Common brands are Molex, Amphenol, and JST. It’s probably a JST or a knock off, but they make a boatload of different products. Are there any numbers on either the socket or plug?