  We are motivated by cone spherical metrics on compact Riemann surfaces of
positive genus to solve a special case of the Hurwitz problem. Precisely
speaking, letting $d,\,g$ and $\ell$ be three positive integers and $\Lambda$
be the following collection of $(\ell+2)$ partitions of a positive integer $d$:
\[(a_1,\cdots, a_p),\,(b_1,\cdots, b_q),\,(m_1+1,1,\cdots,1),\cdots,
(m_{\ell}+1,1,\cdots,1),\] where $(m_1,\cdots, m_{\ell})$ is a partition of
$p+q-2+2g$, we prove that there exists a branched cover from some compact
Riemann surface of genus $g$ to the Riemann sphere ${\Bbb P}^1$ with branch
data $\Lambda$. An analogue for the genus-zero case was found by the first two
authors ({\it Algebra Colloq.} {\bf 27} (2020), no. 2, 231-246), who were
stimulated by such metrics on ${\Bbb P}^1$ and conjectured the veracity of the
above statement there.
