Can a PRF distiguisher invoke the function's algorithm?
The adversary $\mathcal{A}$ is allowed to invoke the algorithm $\mathcal{F}$ (if it is P.P.T.) in any PRF definiton known to me.
We are generally interested in security against every possible P.P.T. algorithm $\mathcal{A}$ and require that for every such algorithm $\mathcal{A}$ it holds that $\mathcal{A}$ can only distinguish from a random function with non-negligible probability.
If $\mathcal{F}$ is a P.P.T. algorithm, there exists an adversary $\mathcal{A}$ that includes the $\mathcal{F}$ functionality. This adversary is able to invoke $\mathcal{F}$ and we require from our PRF, that it is secure against that adversary too.
The same argument holds for parts of the algorithm $\mathcal{F}$.

As far as I understood it, for your example the important question is the following:
Does the adversary $\mathcal{A}$ know $c$?
Again, we need indistinguishability against all adversaries from a PRF, which means that we need indistinguishability even against an adversary that knows this fixed $c$.