Is a PRF applied to a secure MAC also a secure MAC? Suppose I apply a PRF to a secure MAC.  Do I still have a secure MAC?
Yes.

A secure PRF is a secure MAC. A secure MAC of a secure MAC is a secure MAC. Therefore, applying a PRF to a MAC still gives you a MAC.

Depending on the length of the inner MAC and the PRF you may lose security bits, but if they are long enough it works.