Nonnormal finite cover of a higher-genus orientable surface

For $g\ge 2$ and $n\ge 3$, sendwhere $\sigma =(12\cdots n)$ and $\tau =(12)$ in $S_n$, and send all remaining surface generators to the identity. The two commutators cancel, and $\sigma ,\tau$ generate $S_n$. The inverse image of a point stabilizer is a nonnormal subgroup of index $n$, hence defines a connected nonnormal degree-$n$ cover.