= Solution
Choose a local coordinate $w$ with $w(p)=0$. Differentiation at $p$ gives a homomorphism
$$
\lambda:H\to\mathbb C^\times,
\qquad \lambda(h)=h'(p).
$$
It is injective. Indeed, a nonidentity <Möbius transformation> fixing $p$ and having derivative one there is parabolic, hence conjugate to a nonzero translation and of infinite order; no such element can belong to the finite group $H$. Every finite subgroup of $\mathbb C^\times$ is a cyclic group of roots of unity, so $H$ is cyclic. Write $n=|H|$.
The averaged coordinate
$$
z(q)=\frac1n\sum_{h\in H}\lambda(h)^{-1}w(hq)
$$
has derivative one at $p$, so it is a coordinate on a smaller neighbourhood. For $g\in H$, reindexing by $k=hg$ gives
$$
z(gq)=\lambda(g)z(q).
$$
After shrinking to an $H$-invariant disc $U$, the group therefore acts as all rotations $z\mapsto\zeta^jz$, where $\zeta$ is a primitive $n$th root of unity. The invariant coordinate
$$
u=z^n
$$
identifies $H\backslash U$ with a disc and gives it a <Riemann surfaces>[Riemann-surface] chart. In these coordinates the quotient map is exactly
$$
\boxed{z\longmapsto z^n}.
$$
This is the <local cyclic quotient of a Riemann surface>.
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