On Friday, 3 March 2017 18:59:11 UTC, D.M. Procida wrote:
Here are the facts:

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.

Thanks,

Daniele

Disconnect battery and charge with battery charger.
Next morning/day, (ie when the battery has been left for a while not connected to anything) reconnect and try to start car.
If car starts, battery is OK and something in the car is draining the battery or the alternator is not charging the battery.

The object of the exercise being to see if a known to be fully charged battery will hold a charge when it's not connected to anything that could discharge it.