Monogenic polynomials having squarefull discriminant

Given an integer $n \geq 2$, we produce infinitely many new families of monogenic polynomials $f(x)= x^{n-km} (x^k+A)^m+B \in \mathbb Z[x]$ having a powerful and $k$-full discriminant. Further, under certain assumptions, we give the asymptotics for the number of such monogenic polynomials with $B \leq X$.