Hopf Galois structures, skew braces for groups of size $p^{n}q$: The cyclic Sylow subgroup case

Let $n\ge 1$ be an integer, $p$, $q$ be distinct odd primes. Let $G$, $N$ be two groups of order $p^{n}q$ 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.