Grassmann secants, identifiability, and linear systems of tensors

For any irreducible non-degenerate variety $X\subset {\mathbb{P}}^r$, we give a criterion for the $(k,s)$-identifiability of $X$. If $k\le s-1<r$, then the $(k,s)$-identifiability holds for $X$ if and only if the $s$-identifiability holds for the Segre product $Seg({\mathbb{P}}^k\times X)$. Moreover, if the $s$-th secant variety of $X$ is not defective and it does not fill the ambient space, then we can produce a family of pairs $(k,s)$ for which the $(k,s)$-identifiability holds for $X$.