This question may be easy but I could not come up with a proof.
Let $F$ be a hyperbolic surface of finite type (with finitely many boundary and finitely many puncture). Let $\gamma$ be a closed non-simple geodesic. $\gamma$ is not homotopic to a point, a puncture or a boundary. Let $p$ be a self intersection point of $\gamma.$
1) Can there be a simple closed geodesic passing through $p$?
If not then
2) Can there be a closed geodesic passing through $p$ different from $\gamma$?