Let $X$ a connected Riemann surface and $G$ a finite group that acts faithfully and holomorphically on $X$. Further, let $x \in X$ a non-trivially stabilized point (we know these points are discrete), $U$ a neighborhood of $x$, such that $U \cap gU = \emptyset$ for all $g\notin G_x := \{g \in G : g.x=x\}$ and
$$V:= \bigcup_{g\in G_x} gU.$$
With this setup, my lecture notes say that we can assume "without loss of generality" that $V$ is simply connected. If not, we make $U$ smaller till $V$ has no wholes.
I have thought on this for a long time but I still don't know how to prove this easily. I could not find an counterexample either. Probably one knows an "algorithm" that determines $U$ based on $x$ and $G$ such that $V$ is simply connected.