Dad had the same as Cragmaid - all actions written up in book with time of arrival and time of leaving. Sometimes dad was really co-operative and the visit was short but ALL the agreed care was given - sometimes he was awkward and they kindly stayed longer so that all the agreed care was given. He paid the same for the visits.
His morning visit was to get him up, washed/showered, dressed, have breakfast, take meds, leave him in lounge with TV or music and a coffee = 45 mins. The evening visit was shorter = 20 mins to give him meds, toilet, into pjs, teeth out, settle in bed, lights out (it was rarely as easy as that).
Seems to me that the times you have aren't the best times for the break that the care visit is meant to be for YOU. If your mum is still doing her personal care (are you SURE? is she fibbing and the carers do actually need to gently support her?) I seem to remember that bedtimes had been difficult for you. But maybe the visits need to be swapped for other time slots or for mealtimes - to get her lunch and meds and a dinner and meds - to take that responsibility from you.
It seems odd that you have an hour allotted but it's not used - what exactly is it meant to be for? IF the carers are meant to spend time with your mother then they should stay and chat with her or even just sit with her for the given time - to give you a break, so you can safely do whatever you want.
I'd be resistant to give up care if I were you, as you will need it at some point and at the very least your mother is getting used to others coming into the house regularly to help you both - that in itself is often a breakthrough.
Are you sure YOU aren't doing what the care givers are meant to? If so, hard as it is, step back and let them get on with it.
PS Just reread your posts and seen that the first visit is at 1pm = lunchtime? So they do not come to help her in the morning. What are they meant to do then?
Not sure Selinacroft is right - dad's carers had a care plan and the tasks were all agreed so that the carers knew exactly what was expected of them, they were not there to pick up any undone housework - UNLESS that is made part of the agreement.