Symplectic Kloosterman Sums for $\mathrm{Sp}(2n)$ with Powerful Moduli

We prove a non-trivial bound for $\mathrm{Sp}(2n)$ Kloosterman sums of moduli not equal to a prime multiple of the identity. These sums are attached to Siegel modular forms on the group $\mathrm{Sp}(2n)$ and appear in the corresponding Petersson formula. We give an application to equidistribution of coprime symmetric pairs.