If $C_p(M)$ is the cut locus of some $p\in M$ in some Riemannian Manifold $M$, then does there always exist a countable collection of points $\{p_n\}_{m \in \mathbb{N}}$ such that: \begin{equation} \bigcap_{n\in\mathbb{N}} C_{p_n}(M)=\emptyset\text{ ?} \end{equation}
This is true for trivial example of $S_n$, simply take two distinct points but does this hold in general?
2. If not what are the necessary conditions for it to hold?
Yes. For each $p\in M$, the set $U_p = M\smallsetminus C_p$ is a neighborhood of $p$, and these neighborhoods cover $M$. Every open cover of a manifold has a countable subcover.
[Note that the question in your title is different from the one you asked in the text, and has a different answer -- it's certainly possible to find a countable collection of cut loci whose intersection is nonempty.]