Let $f:X\to \mathbf{P}^1$ be a rational function of degree $d\geq 2$ on a curve $X$.
Let $n\geq 2$ be a divisor of $d$. Does there exist a curve $Y$ with a rational function $g:Y\to \mathbf{P}^1$ of degree $n$ such that $f$ factors through $g$
That is, can we factor $f$ in some sense?
Note that by de Franchis' theorem the curve $Y$, if it exists, will almost always be isomorphic to $X$.
Example. The rational function $x\mapsto x^d$ clearly factors as $x\mapsto x^{d/n}$ and $y\mapsto y^n$.