May just need using, all this starting every week and that for 10 min utes does not put the same charge back in as you take out, so it becomes even more drained.
There's a old biddy round here I fitted a C tek to her car as she uses 3 times a week for a 2 mile journey(mile easch way), battery was becoming flat all the time as alarmed etc.
A better battery type to suit would pay dividends, ie a AGM, would be vastly more suited to this kind of work, and would not suffer damage everytime it gets a bit flat.
tabetha