Simple modules over Witt superalgebras from super Weyl modules and $\mf{gl}(m,n)$-modules

For a simple module $P$ over the Weyl super algebra $K_{m,n}^+$ (resp. $K_{m,n}$) and a simple weight module $M$ over the general linear Lie superalgebra $\mf{gl}(m,n)$, the tensor module $P\otimes M$ is a module over the Witt superalgebra $W_{m,n}^+$ (resp. $W_{m,n}$). We obtain the necessary and sufficient conditions for $P\otimes M$ to be simple, and determine all simple subquotient of $P\otimes M$ when it is not simple. All the work leads to completion of some classification problems on the weight representation theory of $W_{m,n}^+$ and $W_{m,n}$.