OK … good news is that the 127 is a nice simple kit. Your clicking will definitely be the air solenoid - perfectly normal, nothing to worry about and more importantly nothing to do with the compressor.
The nice thing about this kit is that it does not use a compressor relay, so you basically only have to worry about one power and one earth wire. Get those two things right and you’ve got a functioning system.
The compressor is essentially turned on/off via the pressure switch mounted on the tank. Here’s a basic diagram of what’s going on and I’ll go into the reference points (1-4) below:
Let me start by saying, that a small cheap multimeter or test light will be your friend here. If you can borrow one of those it will make the job of testing the wires a whole lot easier. OK, so here’s the deal (reference the green points 1-4):
(1) Ensure that your wiring to the battery is A-OK. Check you have 12V coming into your fuse and then back out again the other side
(2) I know you’ve said that you’ve checked the fuse, but test it rather than just look at it. Even just replace it.
(3) HB includes a toggles switch in this kit and normally that’s placed in-line after the fuse but before the compressor. It can however be left out or even placed on the earth line. In any case, check again that you have power running through the switch
(4) Power enters the pressure switch, which is mounted on the tank and then comes back out to go to the compressor. Like before, check that you have power flowing through the pressure switch.
(5) Is where the power coming back out of the DC motor of the compressor goes back to the negative terminal of your battery (i.e. through ground/earth). Most commonly this where the trouble with dodgy wiring will be since people tend to wire it to a bolt or chassis on the car, rather than back to the battery. Depending on the connection used, your earth will tend to corrode and create resistance in the circuit which will cause you a number of issues. Ideally you want a wire leading back to the negative terminal of your battery.
Hope the above makes sense. It’s a very basic circuit and you just have to trace through it to ensure power is able to flow smoothly through all of those components. As I said above, I’d check your earth wire first, then test your compressor by just running two wires direct from the compressor to the battery terminals. The rest is just switches and fuses which are best checked with a meter.
If you’re confused with anything I’ve written above, let me know.