  Let $I\subset R=K[x_1, \ldots, x_n]$ be a square-free monomial ideal,
$\mathfrak{q}$ be a prime monomial ideal in $R$, $h$ be a square-free monomial
in $R$ with $\mathrm{supp}(h) \cap (\mathrm{supp}(\mathfrak{q}) \cup
\mathrm{supp}(I))=\emptyset$, and $L:=I\cap (\mathfrak{q}, h)$. In this paper,
we first focus on the associated primes of powers of $L$ and explore the
normally torsion-freeness of $L$. We also give an application on a comb
inatorial result. Next, we study when a square-free monomial ideal is minimally
not normally torsion-free. Particularly, we introduce a class of square-free
monomial ideals, which are minimally not normally torsion-free.
