There are a lot of reasons for a bad idle but this relates to one specific type of Protégé idle. About 4 months ago my Protégé would make this "pomp" sound at random every 5-10 seconds when at fast idle when first started and then stop. Later the car would almost die when stopped at a traffic light but recover. This would happen at random also. When I would turn the lights on or AC on, the car would shake and vibrate for a short time and then stop and return to normal. All of this would go away for a day an then come back then go away etc. I read the forums, Youtube, and the repair book. I checked everything that was given as a cause. The only things I could find that were not in specs were the PCV valve and a cracked small vacuum hose that goes into the throttle body area. I also put in new spark plugs. Both were repaired. Thought that fixed it because the problem went away but then returned.
Then I remembered a similar situation with a Nissan Quest.
I turned out the Nissan had a slightly clogged fuel injector or two that was not working properly (engine almost twice as large as the Protge, 6 cylinders). The cure for that was:
A fuel cocktail composed of one large bottle of Tectron 3000 mile fuel injector and engine cleaner AND a good dose of Marvel Mystery oil added to the gas.
So I gave 1/2 tank of gas in my Protégé the cocktail and after three days I have no problems at all. The car has more power and idles as smooth as silk. Not once has the car acted up under any circumstances.
The 1.6 is a tiny engine and any amount of clogged injector will cause problems and the computer will compensate before the car shuts off.
I may have been something else in the fuel/pollution control system that was gummed up by ethanol but I think it was ONE injector.
This may not be your problem, but it is cheap and it could save you a lot of trouble and it will not harm your engine.
Good luck.