Cover Supervisor roles are quite new and do vary from school to school, but this is the situation in my school:
I am based in the English and MFL Departments (my degree is in German). For up to 15 hrs a week (average 13) I cover classes when the teachers are absent - planned absences (courses and meetings) and illness. I am allowed to cover 3-5 days staff absence. Usually staff will set work (that can be delivered by a non-specialist and doesn't require "teaching"), but sometimes I have to sort out the cover work as well. I am first on call for cover in my departments, but can be asked to cover across the school - and over two years I have had classes in every single classroom in the school.
The remainder of my time is given to supporting my departments. I do some classroom support and team teaching. I am responsible for displays throughout the departments, minute-taking at meetings, and misc dept admin. I also adminster the two European trips for the MFL department, including taking all the payments, and liaising with parents. I also cover the isolation unit whilst the supervisor has a lunchbreak once a week. I run two extra-curric reading groups, and help with a P4C Club.
Some schools do not give their CS these admin/support roles, but they do more cover than we do.
Some Unions and teachers don't approve of CS at all, but the fact of the matter is most schools cannot afford to pay supply rates for qualified supply teachers to cover short-term absences. In our school CS have been a great success, and are popular with all departments and nearly all staff - including those that were sceptical at first. Most departments are highly supportive and protective of their CS.
For planned absences most staff in my dept will request in advance that I cover their lessons rather than a supply, because they know the students will (mostly) behave and get their work done
I also join in the general staff ethos of the school, with corridor duties, staff meetings, INSET and so on - and being up at 3 am because we've got O#**#D in this week and I can't sleep