Does a Kloosterman sum composed with a rational function exhibit square root cancellation?

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 $t$.

• It seems a result of Fouvry and Michel (infoscience.epfl.ch/record/129095/files/somexp.pdf) combined with a result of Hooley (which they seem to be implicitly claiming without proof extends to rational functions) reduces matters to showing that xR(t)+x^{-1} isn't the composition of bivariate rational function with a linear fractional transformation over the closure of F[T]. – Mark Lewko Jan 10 '18 at 6:49