@Stormwalker: This may be a long shot -it’s unlikely to be your problem - but it’s easy to check even for the mechanically DIS-inclined ;=). It’s worth checking because the fix is potentially free or very cheap. Or more likely, you’ll rule this out as your problem.
Sometime after dark just open the hood, start the engine, and look at the engine(without a flashlight) for small bluish pinpoint flashes of light. DON’T TOUCH ANYTHING! In the dark, any such flashes would be very easy to notice.
If you see any flashes of light, it means the electricity which should be going to the spark plug is somehow escaping from the wire and “grounding out” somewhere that it shouldn’t. It’s high voltage and would make a snap sound, maybe that’s the click you’ve been hearing. This could be a wire that’s come off the spark plug or something related. That would make the engine to run poorly - which you would feel while driving. Use a flashlight to get a better idea of where the light is coming from, so you can find it later, or show it to someone else.
Shut off the engine and if you dare, see if the source of the flash was definitely a wire that you can see out of place. Maybe you can reconnect it later after the engine cools off. If not, the next day stop at a decent looking car repair shop and tell them you suspect you have a spark plug wire fallen off and don’t know how to reconnect it, and would they do that for you. If they are nice at that shop, then you may have found a shop you can try for more work later as needed.
Since you say the noise goes away after a few minutes, this is probably not your problem, but it’s best to rule out the simplest things first so that you don’t spend money fixing things that aren’t broken.
If you haven’t noticed the “mechanics files” at the top of this page, you might use that to locate a local repair business. Good luck!