The vast majority of people go home halfway through the day - only the die hards stay till the end - so if you had a half day-cap, the majority of people would go for it and you would just end up halving your income...
As always, different hunts have dfferent rules, but I think I would be right in saying that most hunts don't allow late starters, except under special circumstances and by permission, and most would expect you to pay the full cap.
As always, the best way to find out is to ring the Secretary of the hunt concerned.
What would constitute a half-day is the question, something we've been toying with lately as we have a one person in particular who wants to pay a half-cap but has in the past said 'goodnight' after 2pm with others who've paid in full!
As for turning up late, I think our MFH would take a pretty dim view of that and would still expect a full cap to be paid!!
All our local Harrier packs start at 1pm as opposed to our 11am start and as far as I am aware you still pay the same amount of cap. It would be hard to police unless the secretary is out following all the time and made sure that the person went home earlier than 1pm!! Sounds like to much hassle to me.
My hunt don't really allow late starters unless by prior permission and you are expected to pay full cap.