We consider and compare several approaches to the construction of the explicit formulas for the ``universal'' off-shell Bethe vectors for the quantum integrable models associated with the super Yangian $Y(\mathfrak{gl}(m|n))$. These formulas are given as the sums over partitions of the sets of the Bethe parameters. We consider the action of monodromy matrix elements onto these Bethe vectors and prove that the set of the Bethe vectors which are certain polynomials of the monodromy matrix elements is invariant under this action. This allows to reduce a problem of calculation the form-factors of the local operators in the corresponding quantum integrable models to the problem of calculation of the scalar product for the off-shell and on-shell Bethe vectors.