1. The car is used once or twice a week, mostly for short trips.
2. Nearly every time we use it, the battery is too dead to turn the engine over and it needs to be charged. In fact it's so dead that even trying will produce nothing but a click and drop the voltage across the electrical system so low that it resets the clock.
3. After 20 minutes or so of charging, it'll start without difficulty (it's only a 1.2l engine).
4. Alternatively, a quick push will do the trick.
5. If it has just recently been on a long drive (in the previous day or so), or has been used once that day already, it will start without problems, though.
6. More than once recently though I've stopped the car for a few moments after driving several miles, and will find that it doesn't start and needs a push to get going again.
7. It's getting worse.
8. The battery is fairly new.
I'm assuming that the problem is the alternator failing to charge the battery properly, but no. 6 is a puzzle to me.