  In this work we study the monodromy group of covers $\varphi \circ \psi$ of
curves \linebreak $\mathcal{Y}\xrightarrow {\quad {\psi}} \mathcal{X}
\xrightarrow {\quad \varphi} \mathbb{P}^{1}$, where $\psi$ is a $q$-fold cyclic
\'etale cover and $\varphi$ is a totally ramified $p$-fold cover, with $p$ and
$q$ different prime numbers with $p$ odd.
  We show that the Galois group $\mathcal{G}$ of the Galois closure
$\mathcal{Z}$ of $\varphi \circ \psi$ is of the form $ \mathcal{G} =
\mathbb{Z}_q^s \rtimes \mathcal{U}$, where $0 \leq s \leq p-1$ and
$\mathcal{U}$ is a simple transitive permutation group of degree $p$. Since the
simple transitive permutation group of prime degree $p$ are known, and we
construct examples of such covers with these Galois groups, the result is very
different from the previously known case when the cover $\varphi$ was assumed
to be cyclic, in which case the Galois group is of the form $ \mathcal{G} =
\mathbb{Z}_q^s \rtimes \mathbb{Z}_p$.
  Furthermore, we are able to characterize the subgroups $\mathcal{H}$ and
$\mathcal{N}$ of $\mathcal{G}$ such that $\mathcal{Y} =
\mathcal{Z}/\mathcal{N}$ and $X = \mathcal{Z}/\mathcal{H}$.
