Let $X$ be a smooth projective irreducible curve over $\mathbb C$. Let $x$ and $y$ be distinct points.

We say that $f$ is totally ramified at a point $P$ if the ramification index of $P$ equals $\deg(f)$.

Does $X$ admit a finite map $f:X\to \mathbb P^1$ which is totally ramified at $x$ and $y$?