Puncture loops on a non-orientable surface

On a connected surface $N$ with negative Euler characteristic, the free homotopy class of a loop obtained by smoothing an intersection of two closed geodesics may wind around a puncture. Chas and Kabiraj showed that this phenomenon does not occur when the surface $N$ is orientable. In this paper, we prove that it occurs when $N$ is non-orientable and both geodesics involved in the smoothing are actually one-sided. In particular, we study a loop obtained by traversing a one-sided closed geodesic and the $m$-th power of another one-sided closed geodesic for odd $m$. Then we show that its free homotopy class may wind aroud a puncture at most two values of $m$. Furthermore, if two such $m$'s exist, they are consecutive odd integers.