On $p$-gonal fields of definition

Let $S$ be a closed Riemann surface of genus $g\ge 2$ and $\phi$ be a conformal automorphism of $S$ of prime order $p$ such that $S/\langle \varphi \rangle$hasgenuszero\mathrm{.}Let${\mathbb K} \leq {\mathbb C}$ be a field of definition of $S$. We provide an argument for the existence of a field extension $\mathbb{F}$ of $\mathbb{K}$, of degree at most $2(p-1)$, for which $S$ is definable by a curve of the form $y^p=F(x)\in \mathbb{F}[x]$, in which case $\phi$ corresponds to $(x,y)\mapsto (x,e^{2\pi i\mathrm{/}p}y)$. If, moreover, $\phi$ is also definable over $\mathbb{K}$, then $\mathbb{F}$ can be chosen to be at most a quadratic extension of $\mathbb{K}$. For $p=2$, that is when $S$ is hyperelliptic and $\phi$ is its hyperelliptic involution, this fact is due to Mestre (for even genus) and Huggins and Lercier-Ritzenthaler-Sijslingit in the case that $\mathrm{A}\mathrm{u}\mathrm{t}(S)\mathrm{/}\phi ⟩$ is non-trivial.