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The "click" as you probably know is the relay opening back up. Failing or badglow plugs can be indicated by a "hard start" condition, but a better indication is when one cylinder is "missing" when the engine is cold, but the engine runs smoothly when warmed up. Your glow plugs can be tested by you very easily if you wont to check it out. If you do not already know I will be happy to help you out.
Is your lights dimming win your grid cycles? Or your volt meter drooping? We know your relay is opening and closing. We know have to find out if the solenoid is working.
So the preheat and the post-heat cycles are working? The lights/volt meter keep dimming even after it starts? If it does not after the engine has been started the post-heat cycle is not working or is not working long enough. It should be on for 10 sec and off 15 sec +/- depending on air temperature . If this is the case it most likely it is the air temperature sensor. It seems your element, relay etc. is working due to the dimming of your lights.