  Let $n\geq 1$ be an integer, $p$, $q$ be distinct odd primes. Let ${G}$, $N$
be two groups of order $p^nq$ with their Sylow-$p$-subgroups being cyclic. We
enumerate the Hopf-Galois structures on a Galois ${G}$-extension, with type
$N$. This also computes the number of skew braces with additive group
isomorphic to $G$ and multiplicative group isomorphic to $N$. Further when
$q<p$, we give a complete classification of the Hopf-Galois structures on
Galois-$G$-extensions.
