Sounds like a three knob control unit behavior, so I will assume this. If so, sounds like two problems, first the short cycling is most likely the "run" capacitor, the silver metal cased one in the control box on the compressor unit. With the breaker off, take a look at the top of this capacitor where the terminals are located and see if the top of the can is still flat or bulging upward. If bulging replace it. The compressor starting initially indicates to me that the start capacitor and potential relay are functioning correctly and the problem starts when the run capacitor fails to provide the required phase shift to keep the compressor motor running.
Regardless of the status of the run capacitor, you are low on refrigerant as mentioned already, and need to get up in the range of 60 or 70 lbs on the low side when not running. Once there use some gauges and a charging chart to get the setting correct when operating.
Pete