Welcome to the forum I hope you enjoy whatever time you care to spend here... You could have a duff battery, or as has been suggested you could have some sort of constant load also already mentioned above. Battery`s have improved in terms of lifespan in more recent years and as such advanced glass mat (AGM) batteries seem to be those fitted to modern machinery as they seem to last longer than old style wet acid batteries, despite their heavy duty cycle in terms of auto stop/start and numerous other on board electronic control systems.
I regularly read of a similar problem to that you are suffering, which as has been stated could either be down to the condition of your battery as the result of it`s age or the way it has been treated over time, i.e. lack of use The charging system could have an issue, or the parasitic losses clock/radio/alarm etc. etc. etc..
Just look at the battery as a container full of water, if when parked it is not filled to the brim and the container leaks due to age or mistreatment, then it will start to loose that which is stored within, and the amount of losses will be dependant on the extent of the volume lost and the time duration before the container is topped up.... Unfortunately in the case of a battery and a car, it requires an engine start which is quite the heaviest load on the battery to be applied before it can be topped up.
My Morgan lives on a CTEK float charger/battery conditioner when not in use, as do many thousands of seldom used cars of all types. These chargers can mask a degrading battery to some extent allowing an ageing battery to struggle on past a time when it might be wise to replace it... Though you will appreciate that for some it can be difficult to determine the real cause behind a fail to start...?
If buying a CTEK or similar be sure to match it to the type of battery you have in your Morgan as AGM and older wet batteries have different requirements.
Sorry if it seems I am insulting your intelligence in my simplistic attempts to help out.
As ever more than happy to be corrected if I have got anything wrong..
