Turbo or N/A? Either way, every time that baby clicks it's shutting off your fuel pump. No wonder the car is acting flakey.
The circuit opening relay is a safety device. It's sole purpose is to interlock fuel pump to engine operation. It's job is to shut the fuel pump down in the event of sudden engine stoppage after a collision.. Course, it also shuts it down anytime you turn the engine off and prevent the pump from running before you start the engine. Looked at another way, it's there to keep the pump from operating when the key is on but the engine is not running. Every EFI car has a scheme to accomplish this. Older Volvos and Audis do it with an RPM relay while Jags use a G switch. Most cars use a switch in the AFM, as does the N/A Supra. Because of the KV sensor on the turbo the ECU handles the task..
Look at your manual. The relay has two coils. I assume you checked both. One coil is initiated by the crank (STA) signal in both N/A and turbo cars. After engine start the second coil is kept energized by a switch in the AFM if a N/A car or by the ECU if a turbo. If you have a N/A car check the switch in the AFM and it's associated wiring. On either car check the relay socket and the B+ EFI wiring to it. On a turbo car check the ECU connector. A loss either B+ power or the ground supplied by the ECU or AFM flow switch will cause the relay to drop out.
You didn't say if the clicking was regular or follows a pattern, such as when the car is trying to run. I doubt the ECU is shutting it down in a regular pattern so I'm guessing you have a N/A car. Every time the engine starts the relay clicks off when you release the key because the AFM problem keeps the second coil from staying on. Thus the car dies. It's a vicious cycle. I'm assuming that's what you mean when you say it's clicking off and on. Lets hope you have a N/A car as it'll be easier to find the problem. On a turbo the ECU keeps the COR latched based on input other than a simple switch in an AFM.
Try jumpering the two wires for the switch at the AFM. If the problem goes away you need to open the AFM and find out why the vane isn't keeping the switch closed. Either that or replace it. If you have a turbo you need to look elsewhere.
Hope that helps....
