I believe the algebraic variant of what you are asking is the property of _subgroup separability_ in groups (which is very well studied). Recall that a group, $G$, is _subgroup separable_ if for any finitely generated group, $\Delta \leq G$ and any $g \notin \Delta$, there exists a finite index subgroup $H \leq G$ such that $g \notin \Delta H$.

You cannot expect to generalize subgroup separability of free groups to arbitrary free products of freely indecomposable groups, as subgroup separability implies _residual finiteness_ (and any infinite simple group is not residually finite). However, free groups and, more generally, surface groups are known to be subgroup separable. In fact, any group that contains a subgroup of finite index that is subgroup separable is subgroup separable. For the proofs of the aforementioned facts see Peter Scott's _Subgroups of surface groups are almost geometric_ from JLMS, 1978. The main ideas in this paper were greatly generalized: the modern form of this is now called the _canonical completion_ of Haglund and Wise. See Section 6 in _Special Cube Complexes_ from GAFA, 2008.