If the compressor runs without cutting off every few seconds then the low pressure switch wire rig job is working.
If the A/C system is cooling anything in its system, then you will see condensation on the accumulator ( the black canister ) and it will be very cold, sometimes even freezing the condensation on it. If your system is making the accumulator cold under the hood but you are still getting hot air in the cab, then your hot / cold air damper is not working. It is stuck on hot air.
From the picture you posted, I am very concerned about the corrosion on the wires on that plug. That is probably why your mechanic told you that you may have to jiggle that wire.
If I were you I would fix the wires on that plug so they aren't corroded and replace that low pressure switch. You need that switch to work properly because that switch tells your compressor to stop working if the refrigerant level gets too low. With it rigged up with that wire, if you have a slow leak and the refrigerant gets low, you will burn up your new compressor.
Also, no offense intended, but I would find a new mechanic. If he cared about your vehicle and the money you spent on it, he would have told you what I just did in the sentence above.