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Let $L/K$ be a $G$-Galois extension of fields with an $H$-Hopf Galoisstructure of type $N$. We study the ratio $GC(G, N)$, which is the number ofintermediate fields $E$ with $K \subseteq E \subseteq L$ that are in the imageof the Galois correspondence for the $H$-Hopf Galois structure on $L/K$,divided by the number of intermediate fields. By Galois descent, $L \otimes_K H= LN$ where $N$ is a $G$-invariant regular subgroup of $\mathrm{Perm}(G)$, andthen $GC(G, N)$ is the number of $G$-invariant subgroups of $N$, divided by thenumber of subgroups of $G$. We look at the Galois correspondence ratio for aHopf Galois structure by translating the problem into counting certainsubgroups of the corresponding skew brace. We look at skew braces arising fromfinite radical algebras $A$ and from Zappa-Sz\'ep products of finite groups,and in particular when $A^3 = 0$ or the Zappa-Sz\'ep product is a semidirectproduct, in which cases the corresponding skew brace is a bi-skew brace, thatis, a set $G$ with two group operations $\circ$ and $\star$ in such a way that$G$ is a skew brace with either group structure acting as the additive group ofthe skew brace. We obtain the Galois correspondence ratio for several examples.In particular, if $(G, \circ, \star)$ is a bi-skew brace of squarefree order$2m$ where $(G, \circ) \cong Z_{2m}$ is cyclic and $(G, \star) = D_m$ isdihedral, then for large $m$, $GC(Z_{2m},D_m), $ is close to 1/2 while $GC(D_m,Z_{2m})$ is near 0.

On the Galois correspondence for Hopf Galois structures arising from finite radical algebras and Zappa-Sz\'ep products