1998 Honda Civic overheating issue?

Ok, so I recently had my radiator replaced and more recently my ac compressor, clutch, relay, and professionally recharged. I even replaced my blower motor and all of my belts.
Lately I have been having issues where when I idle, my ac compressor clutch clicks on and off every 2 seconds or so. Obviously, my ac warms up after. But then my temp gauge on my dash starts to show it's overheating. I was wondering What the issue could be. Maybe my battery isn't suppling good power to my regulator fan relay and also my compressor? My battery is only 2-3 months old so I'm thinking that might not be the issue… Anyone know where I should look?

You have 2 problems going on here, first is your water pump going out OR your thermostat is stuck closed. And second, when they replaced your a/c compressor they left it with a slow leak, go back and make them find/fix the leak.
When your sitting at a red light and it starts to over heat, turn on your heater to cool it off some.Roll down your windows if you have to.
When you hear a a/c compressor kicking in/out like that its low on freon.

THEIRS A GOOD CHANCE ITS THE SCHRADER VALVE LEAKING SINCE THEY JUST PUT NEW FREON IN IT.

So your engine is overheating well did you check the water pump or thermostat? May need to make sure the coolant system is bled out of air pockets. Make sure your main engine fan comes on after the car reaches operating temp

Check the radiator main fan relay. They go bad and won't turn on the fan, but the ac fan relay might run its fan.

Your AC system still has a leak and you are once again low on 134a refrigerant.
Have a "block test" done to find out if the head gasket is bad.
I'd kick that mechanic in the but for not advising you to buy a different vehicle. You put too much money into an old POS that wasn't worth it.