Let $p$ be a prime and consider the field $\mathbb{F}_p$. Fix $f\in\mathbb{F}_p[X]$ a polynomial of degree $d\ge 2$. Define $$ K(x,y)=\frac{1}{\sqrt{p}}\sum_{z\in\mathbb{F}_p}e_p(xz+yf(z)), $$ where $e_p(t):=e^{2\pi i \frac{t}{p}}$.

It is well-known that if $p\nmid d$, then $|K|$ is bounded by $1$ (up to some constant depending on $d$).

Now consider a sum of products for fixed $t\neq 0$ and $y\neq y'$: $$ I=\sum_{x\in\mathbb{F}_p}K(x,y)\overline{K(x-t,y+t)}\overline{K(x,y')}K(x-t,y'+t). $$ My question is: for what $f\in\mathbb{F}_p[X]$ can we bound $|I|$ by $\sqrt{p}$ (up to a multiplicative constant, of course), or perhaps a little worse: $p^{\delta}$ ($\delta<1$). When $f$ is quadratic, then $K$ is a Gauss sum that can be evaluated explicitly, and I can calculate $I$ and answer my question affirmatively. I don't know what happens for general $f$.