Let $G$ be a HNN extension of the form $\langle s_{1},s_{2},\dots,s_{n},t \mid R, t^{-1}s_{1}t=s_{2}\rangle$, where $R$ is a set of relations in the alphabet $\{s_{2},\dots,s_{n}\}$.

I want to find a finite index subgroup of $G$ substituting $s_{1}$ by a power $s_{1}^m$ and such that $t$ only acts on $s_{1}^m$. I can't take $\langle s_{1}^m,s_{2},\dots,s_{n},t \mid R, t^{-1}s_{1}^m t=s_{2}^m\rangle$ because I would need to add conjugates of $s_{1}^m$ in order to have a finite index subgroup.

I have tried to visualize the problem with Bass-Serre theory, but I can't convince myself that $t$ doesn't touch any other element.

Can someone help me, please?