Residual separability of subgroups in free products with amalgamated subgroup of finite index

Let $P$ be the free product of groups $A$ and $B$ with amalgamated subgroup $H$, where $H$ is a proper subgroup of finite index in $A$ and $B$. We assume that the groups $A$ and $B$ satisfy a nontrivial identity and for each natural $n$ the number of all subgroups of index $n$ in $A$ and $B$ is finite. We prove that all cyclic subgroups in $P$ are residually separable if and only if $P$ is residually finite and all cyclic subgroups in $H$ are residually separable; and all finitely generated subgroups in $P$ are residually separable if and only if $P$ is residually finite and all subgroups that are the intersections of $H$ with finitely generated subgroups of $P$ are finitely separable in $H$.