On the number of prime factors with a given multiplicity over h-free and h-full numbers

Let $k$ and $n$ be natural numbers. Let $\omega_k(n)$ denote the number of distinct prime factors of $n$ with multiplicity $k$ as studied by Elma and the third author. We obtain asymptotic estimates for the first and the second moments of $\omega_k(n)$ when restricted to the set of $h$-free and $h$-full numbers. We prove that $\omega_1(n)$ has normal order $\log \log n$ over $h$-free numbers, $\omega_h(n)$ has normal order $\log \log n$ over $h$-full numbers, and both of them satisfy the Erd\H{o}s-Kac Theorem. Finally, we prove that the functions $\omega_k(n)$ with $1 < k < h$ do not have normal order over $h$-free numbers and $\omega_k(n)$ with $k > h$ do not have normal order over $h$-full numbers.