FFLV bases for covariant representations of $\mathfrak{g}\mathfrak{l}(m\mathrm{\mid }n)$

We study the PBW filtration on covariant representations for the Lie superalgebra $\mathfrak{g}\mathfrak{l}(m\mathrm{\mid }n)$. We prove for all covariant weights of the form $(\lambda \mathrm{\mid }\mu ,0^{n-1})$, that there exists a lattice polytope such that the lattice points of this polytope parametrize a basis of the corresponding associated graded space. As a consequence, we obtain degenerations of partial flag supervarieties for the supergroup $GL(m\mathrm{\mid }n)$ into toric supervarieties.