Let $X$ be a hyperelliptic curve of genus $g>1$.

For which $g$ does $X$ admit a map $X\to \mathbb P^1$ of degree $3$?

I think a genus two curve $X$ admits a map of degree $3$.

Proof: Pick $P$ and $Q$ such that $P+Q$ is not linearly equivalent to $K_X$. Now compute $h^0(P+Q) = 1$ using Riemann-Roch and the fact that $h^0(K_X-P-Q) =0$. For any point $R$ we have that $h^0(P+Q+R) = 2$, so there exists a function of degree $3$ on $X$ (whose divisor of poles is $P+Q+R$). QED

This computation doesn't work in higher genus making me suspect that for $g>2$ a hyperelliptic curve does not admit a degree 3 map.