Is there any special name for the processes like PV^a(a need not be the ratio of specific heat capacities at constant pressure and volume)=constant? How can we calculate molar heat capacity for these types of processes?
To answer part one of your question, they are called polytropic processes.
I don't know a lot about calculating molar heat capacity. Probably best to do some research on that yourself to see what you can find (I'm sure knowing how to name the process will help).