  In this paper, we construct a homomorphism from the affine super Yangian
$Y_{\ve_1,\ve_2}(\widehat{\mathfrak{sl}}(m|n))$ to the universal enveloping
algebra of the rectangular $W$-superalgebra
$W^{k}(\mathfrak{gl}(ml|nl),(l^{(m|n)}))$. We also show that the image of this
homomorphism is dense provided that $k+(m-n)(l-1)\neq0$.
