For $m,n,c \in \mathbb{N}$, let $S(m,n;c)$ denote the Kloosterman sum $$ S(m,n;c) := \sum_{\substack{1 \leq a < c \\ \gcd(a,c) = 1}} e \left( \frac{ma + n\overline{a}}{c} \right) $$ where $e(n) = e^{2 \pi i n}$ and $\overline{a}$ denotes the multiplicative inverse of $a \bmod c$.

In my research, involving producing a subconvexity bound for automorphic L-functions, I've recently come across a twisted shifted sum of Kloosterman sums. Let $\chi(\cdot)$ denote a Dirichlet character mod a prime $p$. Then I'm looking at $$ F(a, h, \chi) = \sum_{b \bmod p} \chi(b) S(a, b; p) S(a, b + h; p) \tag{1}.$$

I've never seen sums like this appear, but it looks pretty complicated. A first thing to consider might be an upper bound. We can produce a trivial upper bound using the Weil bound for Kloosterman sums, which indicates that $(1)$ is bounded above by $p^{2 + \epsilon}$, independently of $a,h, \chi$. But I think we should expect much smaller, at the most $p^{3/2 + \epsilon}$.

So I am wondering if someone has considered sums similar to $(1)$. I would also be interested in considerations of the similar but simpler sums $$ \begin{align} F(a, 1, 1) &= \sum_{b \bmod p} S(a, b; p) S(a, b + 1; p) \tag{2} \\ F(a, 1, \chi) &= \sum_{b \bmod p} \chi(b) S(a, b; p) S(a, b + 1; p). \tag{3} \end{align}$$