Suppose one knows an $(m, n)$ matrix $A$ along with some rank-revealing factorization (let's say the economy SVD for conreteness), $A = U S V^H$, with $S$ an $(r, r)$ matrix where $r$ is the numerical rank of $A$. Various algorithms exist to efficiently obtain the decomposition of the low-rank update $A + B C^H$, where $B$ and $C$ have small rank.
Suppose one seeks an update which is not low rank, but only because $B$ and $C$ take the form e.g. $B = I \otimes b$, where $b$ is a $(d, d)$ matrix with $d << $ any of $r, m, n$, and $I$ is the identity of dimension $m-d$. Can anything be said about the update in this case?