In some recent work, the following strange-looking exponential sum arose: $$ \sum_k \sum_r \sum_s e\bigg( \frac{r \bar s^{(r)} \bar k^{(r^2+s^2)}}{r^2+s^2} \bigg). $$ Here $e(x) = e^{2\pi i x}$ as usual, and $\bar a^{(b)}$ denotes the inverse of $a$ modulo $b$. (For simplicity, let's think of the three sums as being over intervals of positive integers, excluding any integers for which the modular inverses aren't defined.)
Related versions of the sum are something like $$ \sum_k \sum_r \sum_s e\bigg( \frac{\bar s^{(r)} \bar k^{(r^2+s^2)}}{r} \bigg) $$ and $$ \sum_k \sum_r \sum_s e\bigg( \frac{\bar s^{(r)} \bar (r^2+s^2)^{(k)}(r^2+s^2)}{rk} \bigg). $$ In all these cases, we have inverses to two different moduli running around.
Is there anything in the literature that addresses these multi-modulus (Ramanujan or) Kloosterman sums? Or is there an approach for deriving estimates for them more directly from Weyl's/Deligne's estimates?