It is well known that the set of branched covers $X\to \mathbf{P}^1(\mathbf{C})$ of bounded degree and given branch locus in $\mathbb P^1(\mathbb C)$ is finite (up to isomorphism).
Let $X$ be a curve of genus at least two. Do there exist an integer $n\geq 2$ and a finite set of finite places $R$ of $X$ such that the set of degree $n$ finite morphisms $X\to \mathbf{P}^1(\mathbf{C})$ whose ramification locus is contained in $R$ is infinite, up to equivalence?
If $f$ and $g$ are finite morphisms from $X$ to $\mathbb P^1$, they are equivalent if $f$ equals $g$ up to an automorphism of $\mathbb P^1$.