You may have the A/C overcharged, and when the clutch kicks in and spins the compressor, the head or "high side" pressure sky rockets and causes the high pressure switch to open, cutting off the voltage to the compressor clutch.
I don't think it is your compressor or clutch, as when you jump it, it works.
It's also possible you have too low of refrigerant in the system, causing the low pressure switch to "open" killing the voltage to the clutch.
Or you have a bad connection somewhere in the wiring to the clutch.
With the compressor at rest, does the suction or "low pressure" side of your gauge max out?
It should if there is enough charge, as the clutch engages and spins the compressor, the the pressure on the low side will reduce as it's a "suction"
If the gauge is say at 60PSI for a random example, at rest, there may be enough pressure to engage the compressor clutch, but as soon as the compressor runs and "pulls down" the pressure below the "cut out" pressure the low pressure switch is designed to shut off the clutch at, your clutch coil's voltage shuts off, and you have no compressor rotation.
My friend thought his car's A/C charge was correct as he put on the gauge, little cheapy gauge that came with a can of R134-a refigerant, it said ok.
So he gave up.
What he did not understand is that "ok" area on the gauge means when the compressor is running, not at rest.
I opened the button to release the refrigerant into the car's A/C as he started the car and turned A/C on, then the compressor kicked in from the extra pressure of the refrigerant coming into the system, when I let off the suction side would pull down real low and cause the clutch to kick out stopping the compressor.
I kept squeezing until the A/C would stay on by itself and the pressure got to the "ok" area on the cheapy gauge, and within 30 seconds or so he said "hey I feel cold air!"
Maybe you already know this, maybe not?
If so, no offense meant, I'm a HVAC and refrigeration technician, not a auto A/C mechanic, my terms low and high pressure cut out switches are for the refrigeration and building A/C systems.
I think on automotive, it is one switch that does both.
So, without jumping the compressor, what does your suction or low side gauge read( the compressor not being spun ), it should be pegged out to max or very close, if not, the car is most likely low on refrigerant.
