Let $X$ be a compact connected Riemann surface, and let $S\subset X$ be a finite subset.
Does there exist a morphism $f:X\to \mathbf{P}^1(\mathbf{C})$ which is unramified at the points of $S$?
I'm interested in the case where $X$ is of genus at least $2$. (The genus zero case is trivial: take $f$ to be the identity.)
The answer is trivial when $S$ is empty. (Any morphism $f:X\to \mathbf{P}^1(\mathbf{C})$ will do.)
Let $h:X\to \mathbf{P}^1(\mathbf{C})$ be a morphism with ramification locus $R(h)$. Then, if $S\subset X\backslash R(h)$, the answer is yes.
How effective can our answer be? That is, suppose that there exists such an $f$. Then, can we bound its degree?
The title is a special case of the above question: take $S=\{\textrm{pt}\}$.