@Nathaniel Johnston already answered in comments. On a more general note, detecting whether a matrix is a sum of $r$ Kronecker products is essentially the same problem (up to a permutation of the entries) as determining whether a matrix has rank $r$.

Indeed, given a matrix $M\in\mathbb{F}^{am \times bn}$ (divided into blocks $M_{ij}$ each of size $m\times n$), rearrange its entries to form a matrix $N\in\mathbb{F}^{mn \times ab}$ of such that each column of $N$ is the vectorization of a block $M_{ij}$; then a rank-$r$ decomposition of $N = u_1 v_1^T + \dots + u_r v_r^T$ corresponds to a decomposition of $M$ as sum of $r$ Kronecker products $M = \operatorname{vec}^{-1}(v_1) \otimes \operatorname{vec}^{-1}(u_1) + \dots + M = \operatorname{vec}^{-1}(v_r) \otimes \operatorname{vec}^{-1}(u_r)$.

@Nathaniel Johnston already answered in comments. On a more general note, detecting whether a matrix is a sum of $r$ Kronecker products is essentially the same problem (up to a permutation of the entries) as determining whether a matrix has rank $r$.

Indeed, given a matrix $M\in\mathbb{F}^{am \times bn}$ (divided into blocks $M_{ij}$ each of size $m\times n$), rearrange its entries to form a matrix $N\in\mathbb{F}^{mn \times ab}$ such that each column of $N$ is the vectorization of a block $M_{ij}$; then a rank-$r$ decomposition of $N = u_1 v_1^T + \dots + u_r v_r^T$ corresponds to a decomposition of $M$ as sum of $r$ Kronecker products $M = \operatorname{vec}^{-1}(v_1) \otimes \operatorname{vec}^{-1}(u_1) + \dots + M = \operatorname{vec}^{-1}(v_r) \otimes \operatorname{vec}^{-1}(u_r)$.

@Nathaniel Johnston already answered in comments. On a more general note, detecting whether a matrix $M = (M_{ij})$ is a sum of $r$ Kronecker products is essentially the same problem (up to a permutation of the entries) as determining whether a matrix has rank $r$.

Indeed, given a matrix $M\in\mathbb{F}^{am \times bn}$ (divided into blocks $M_{ij}$ each of size $m\times n$), rearrange its entries to form a matrix $N\in\mathbb{mn \times ab}$ of such that each column of $N$ is the vectorization of a block $M_{ij}$; then a rank-$r$ decomposition of $N = u_1 v_1^T + \dots + u_r v_r^T$ corresponds to a decomposition of $M$ as sum of $r$ Kronecker products $M = \operatorname{vec}^{-1}(v_1) \otimes \operatorname{vec}^{-1}(u_1) + \dots + M = \operatorname{vec}^{-1}(v_r) \otimes \operatorname{vec}^{-1}(u_r)$.

@Nathaniel Johnston already answered in comments. On a more general note, detecting whether a matrix is a sum of $r$ Kronecker products is essentially the same problem (up to a permutation of the entries) as determining whether a matrix has rank $r$.

Indeed, given a matrix $M\in\mathbb{F}^{am \times bn}$ (divided into blocks $M_{ij}$ each of size $m\times n$), rearrange its entries to form a matrix $N\in\mathbb{F}^{mn \times ab}$ of such that each column of $N$ is the vectorization of a block $M_{ij}$; then a rank-$r$ decomposition of $N = u_1 v_1^T + \dots + u_r v_r^T$ corresponds to a decomposition of $M$ as sum of $r$ Kronecker products $M = \operatorname{vec}^{-1}(v_1) \otimes \operatorname{vec}^{-1}(u_1) + \dots + M = \operatorname{vec}^{-1}(v_r) \otimes \operatorname{vec}^{-1}(u_r)$.