I encounter a hyper-Kloosterman sum which needs some help from the experts here:

For any integers $q,s \in \mathbb{N}^+$(which may not be necessarily co-prime with each other), how to bound the sum: $$\sideset{_{}^{}}{^{\ast}_{}}\sum _{\substack{x,y \bmod {q}}\\ (y+s,q)=1} \, e\left (\frac{x+\overline{xy} +\overline{y+s}+y }{q}\right)?$$

It is conjectured that the upper-bound should be $O(q^{1+\varepsilon})$, but it looks very challenging to prove this for me.

Note: It is known that one has the estimate $\sideset{_{}^{}}{^{\ast}_{}}\sum _{x,y \bmod {q}} e\left (({x+\overline{xy} +\overline{y}+y })/{q}\right)\ll q^{1+\varepsilon}$, but I don't know how to estimate the sum above with the parameter $s$ involved.

Let $$f (x_1,x_2)=x_1+\overline{x_1x_2} +\overline{x_2+s}+x_2\in \mathbb{F}_q[x_1,x_2,\overline{x_1x_2}]$$ be a Laurent polynomial. One the other hand, one can consider the Newton polyhedron $\Delta(f)$ and verify that that the Laurent polynomial $f$ is non-degenerate by, e.g., Adolphson-Sperber's works. However, I'am really not familiar with algebraic geometry.

I searched Friedlander-Iwaniec's paper "Incomplete Kloosterman Sums and a Divisor Problem" (see https://sci-hub.wf/10.2307/1971175), and Yitang Zhang's paper "Bounded gaps between primes" (see https://annals.math.princeton.edu/wp-content/uploads/annals-v179-n3-p07-s.pdf). It seems that there is no a direct record regarding this type of double exponential sums.

So, if any expert here knows some knowledge on this question, please show some guides or the corresponding references, many many thanks.

Thanks in advance!