Single click sound after ignition is turned off

What is your question?

I inherited a 2000 Elantra years ago and still use it as a third car. I noticed a few years ago that sometimes it would randomly not start, and the battery then needed to be charged from my high-amp charger. Then it would be fine for awhile. I then noticed that right after I switched the car off by turning the ignition key, I would hear a single click sound (and it is not that thing where there are multiple clicks during cooling contraction). -- If I waited till after the click to remove the key, the car always started and the random problem didn't occur. But, if I took the key out quickly, before the click sound, sometimes the car would need to be charged externally to start. -- So what is that click sound? If I take the key out before the click sound, am I leaving some part of the electrical system operating, draining the battery? It's done this through several different batteries over the years, so I am pretty sure it's not a defective battery issue. Alternator???? Advice?