Electric air pump doesn't seem to be working

What is the name of the pump, and what system is it? Why does air tube go through firewall, and what does it control?

I know this is old post but only link I found to my problem.
I jumped green wire. Pump runs but only when I do that.
What does that mean? What needs to be replaced?
Also I left it running and found it does pump air but guessing leak somewhere.
What can I add to line that will escape leak hole and show itself without damaging the system?

I can hear where the leak is but have no luck stopping it.
The hose going into black canister at compressor leaks. When I remove hose and plug with finger it holds pressure. I hook up hose and I can hear air escaping. Have to create a small reservoir to hold water so I can submerge the area. Spraying it didnt work but I can easily hear it and know where it is.
The line is so small in diameter. Is there a way to cut it and put new end on it for better fit? It is a plastic air line with plastic compression fitting on end. Two small washers on backside of compression end.