I think it depends on whether both New Adventure and Cardiff Bus runs are contracted by Cardiff County Council. If, for example, Cardiff Bus runs it commercially then I think it won't be accepted as it's simply another operator, however if all runs are then it's quite possible they will be accepted because it's essentially a council ticket.
I'm not 100% sure on how it works, I don't know if the ticket sales go directly to the council and then the council pay the operator a fixed fee for each journey or if part of it goes to the council and part to operator. Or maybe even the ticket sales go to the operator and the council "top it up".