Denote the classical Kloosterman and Salié sums, respectively, as $KL(a,b) = \sum_{r \in F_*} e(ar+\frac{b}{r})$ and $SL(a,b) =\sum_{r \in F_*} \chi(r) e(ar+\frac{b}{r})$, where $\chi(\cdot)$ is the quadratic character. It follows from Weil's proof of the Riemann hypothesis for curves (or more elementary considerations in the case of $SL(a,b)$) that for $a,b \neq 0$ that $|K(a,b)|, |SL(a,b)| \lesssim |F|^{1/2}$.

Informally, I'd like to know if we can prove square root cancellation in the composition of a Kloosterman (respectively, Salié) sum with a rational function. In particular:

Let $R(t)$ be a rational function in $t$. For $b \neq 0$ can one prove $$ |\sum_{t \in F}'KL(R(t),b)|\lesssim |F|$$ and/or $$|\sum_{t \in F}'SL(R(t),b) \lesssim |F|$$ where $\sum'$ denotes the exclusion of the poles of $R(t)$. Of course the implicit constant should depend on $R$ but be independent of the field?

I'm particularly interested in the special case when the numerator and denominator of $R$ have degree $2$, but would be very interested in the general case as well.

There's been a lot of work by Bombeiri, Deligne, Hooley, Katz, Sperber, and others on deducing multivariate exponential sum estimates from Deligne's theorems, but many of these theorems are hard to parse for someone who isn't an expert in algebraic geometry. I'd also be interested in the above sums twisted by a multiplicative character in $t$.