Question. Suppose $\kappa$ is a supercompact cardinal and $\lambda > \kappa$ is measurable (or even larger large cardinal if necessary). Is there a set generic extension of the universe in which $\kappa$ remains supercompact, $\lambda$ is preserved and $cf(\lambda)=\omega?$
Remark 1. By Gitik-Shelah indestructibility result, if supercompact cardinal is replaced with strong cardinal, then the answer is yes.
Remark 2. If we require that the forcing preserves $\lambda^+,$ then the answer is no, as it is shown by Yair.
Remark 3. In A note on sequences witnessing singularity - following Magidor-Sinapova, Gitik has conjectured the following:
Conjecture. Suppose that
$V ⊆ W$ models of ZFC with same ordinals,
$κ$ is a regular cardinal in $V$,
$cof(κ) = ω$ in $W$,
$V, W$ agree about a final segment of cardinals.
Then there is a subclass $V′$ of $V$ which is a model of $ZFC$, agree with $V$ about a final segment of cardinals, and there is a sequence witnessing singularity of $κ$ (in $W$) which is generic over $V′$ for either Namba, Woodin tower or Prikry type forcing.
Assuming this conjecture, it seems quite plausible that the answer to the question might be no in general.
Edition. I realized that the question has connection with recent work of Woodin:
Theorem. Assume $\kappa$ is an extendible cardinal. If Woodin's $HOD$-conjecture holds, then we can not change the cofinality of some large cardinal $\lambda > \kappa$, preserving the supercompactness of $\kappa,$ by set forcing without collapsing $\lambda.$