A parity learning question
The discussion in the comments below indicades that I have misunderstood the question.  My answer is premised on the Oracle taking no input and returning $(x, f(x))$ where $x \sim p$ or $x \sim q$, depending on $f \in F$. This is apparently not what's being asked.



Because the target distribution is fixed for every target $f^* \in F$, the PAC-sample upper bound applies (this follows from the fact that the target distribution for this bound can even completely depend on $f^*$). Hence, 
$$
m \le \tilde{O}\left(\frac{1}{\epsilon}\left(\mathrm{VC}(F) + \log(1/\delta) \right) \right)
$$
examples should suffice to find a hypothesis of error $\le \epsilon$ w.p. $\ge 1-\delta$. Note -- after seeing these examples, one needs to find a consistent hypothesis from $F$, and this may not be tractable.

On the other hand, one can get a nearly matching lower bound even for the case of $p=q=U$, the uniform distribution, where $m \ge \Omega(\mathrm{VC}(F))$ examples are still required (this can be improved slightly).

The variational distance between $p$ and $q$, as well as $k$ may play a role in the small gap between these bounds, but I doubt it.