Blocking sets and power residues modulo integers with bounded number of prime factors

Let $q$ be an odd prime and $k$ be a natural number. We show that a finite set $S$ of integers that does not contain any perfect $q$th power, contains a $q$th power residue modulo almost every natural number $N$ with at most $k$ prime factors if and only if $S$ corresponds to a $k$-blocking set of $\mathrm{PG}(\mathbb F_{q}^{n})$. Here, $n$ is the number of distinct primes that divide the $q$-free parts of elements of $S$. Consequently, this geometric connection enables us to utilize methods from Galois geometry to derive lower bounds for the cardinalities of such sets $S$ and to completely characterize such $S$ of the smallest and second smallest cardinalities. Furthermore, the property of a finite set of integers of containing a $q$th power residue modulo almost every integer $N$ with at most $k$ prime factors is invariant under the action of the projective general linear group $\mathrm{PGL}(n,q)$.