= Solution
For a nonconstant <holomorphic map> between <compact Riemann surfaces>, choose local coordinates making the map $z\mapsto z^{e_p}$. Its local degree is $e_p$, and its ramification excess is $e_p-1$; branching order conventions use either of these quantities, so both are specified below. The degree $d$ is the number of inverse images of any regular value, or the sum of local degrees in any fiber. The <Riemann-Hurwitz formula> is $2g_X-2=d(2g_Y-2)+\sum_p(e_p-1)$.
The projective closure is $S^m-T^m-Z^m=0$. Its three partial derivatives cannot vanish simultaneously at any projective point, so it is nonsingular and inherits the complex structure of a smooth projective curve. The affine polynomial is irreducible by Eisenstein applied to $t^m-(s^m-1)$ at the prime $s-1$ in $\mathbb C[s]$, so the smooth projective curve is connected. At $Z=0$ its points are $[\zeta:1:0]$, where $\zeta^m=1$; these are exactly the $m$ points added at infinity.
The extension of the coordinate projection is $F=[S:Z]$. There is no point on the curve with $S=Z=0$, so this gives a <holomorphic map> everywhere. More explicitly, in the chart $T=1$ put $\sigma=S/T$, $\tau=Z/T$. Near an infinity point, $\sigma^m=1+\tau^m$ and $\sigma(0)=\zeta\ne0$. Thus $\tau$ is a local coordinate and $1/F=\tau/\sigma$ has nonzero derivative $1/\zeta$ at zero. Each infinity point is unramified, with local degree one.
For a generic finite $s$, there are $m$ distinct solutions of $t^m=s^m-1$, so $\deg F=m$. Ramification occurs exactly at $t=0$, $s^m=1$. At each such point, $t$ is a local coordinate and $s-s_0=t^m/(m s_0^{m-1})+O(t^{2m})$, giving local degree $m$ and ramification excess $m-1$. There are no other critical points, since away from $t=0$ the projection $s$ is a local coordinate. Therefore
$$
2g-2=-2m+m(m-1),\qquad\boxed{g=\frac{(m-1)(m-2)}2.}
$$
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