Let $f:X\longrightarrow \mathbf{P}^1$ be a finite morphism of schemes. If $f$ is etale and $X$ is connected, we can show that $f$ is an isomorphism. That is, the projective line is simply connected. I would like to know if something more general can be said about $f$ if it is not assumed to be etale.

In this generality, it seems quite hard. But what if we assume $X$ to be also the projective line for example? Is it true that $f$ is then of the form $[x:y]\mapsto [x^n:y^n]$ in some sense?