A twisted Kloosterman sum is a character sum of the form $$S(\chi, \psi, \eta)=\sum_{t\in (\mathbb{F}_q)^{\times}} \chi(t) \psi(t) \eta(t^{-1}).$$ Here, $\chi$ is a multiplitive character of $(\mathbb{F}_q)^{\times}$ and $\psi, \eta $ are additive characters. I have two questions:
Why were people so interested in bounding these types of character sums? For example, I know a lot of work went in to actually proving the bound for $|S(\chi, \psi, \eta)|$.
Do sums of twisted Kloosterman sums appear in any relevant context? For instance a sum $$\sum_{\chi\in \widehat{\mathbb{F}_q^{\ast}}}S(\chi, \psi, \eta).$$