I have a Cooper 250w MV Nema that will only light up for a minute before turning off. I know it’s not the bulb or the photocell because I tried 2 brand new GE mercury bulbs in it and no light was on the photocell. It makes 3 clicking sounds before going out. I also didnt find any loose connections. So it must be something with the ballast? Any thoughts???

Have zou tried a shorting cap (or a bypass jumper wire) instead of the photocell?
The clicking sound may point to the photocell switching. The "no light" is highly relative term, photocells use to suffice with rather minimum light and because you are messing around the fixture, the cell may get enough light reflected from yourself...
Otherwise it could be some intermittent contact somewhere getting interrupted once the thing warms up a bit.

Otherwise take a multimeter (and/or a pair of 230V incandescents in series, as there could be 350V on the secondary) and probe the voltage on the lamp, cap and primary just after the clicking shuts it down, it should point to something.

that ballast is HX just photo cell , ballast, and lamp
 I would check all your connection first. 
 use a shorting cap.  If not try bypassiing the photo cell
by putting black wire onto red wire going to the ballast
for testing.  now time test the ballast with a known working lamp. if still giving you problems it time send ballast to a motor rewind  shop have it rewound.