estimate for a sum of products of Weil's sum 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 $f$.
 A: Here is a different approach than Denis Chaperon de Lauzières's.
Opening everything and using orthogonality of characters to remove the $x$ variable, we see that your sum is $1/p$ times $$\sum_{z_1,z_2,z_3,z_4 \in \mathbb F_p, z_1-z_2-z_3+z_4 =0 } e_p \left( y f(z_1) + t z_2 - (y+t) f(z_2)  - y' f(z_3)  -t z_4 + (y'+t) f(z_4) \right)$$
This is an exponential sum of a $3$-variable polynomial. Deligne's theorem (Theorem 8.4 of Weil I) shows that this is $O(p^{3/2})$ (with explicit constant $(d-1)^{3}$) as long as $d$ is prime to $p$ and the leading term of the polynomial inside the $e_p$ defines a smooth hypersurface.
The leading term looks like $$y z_1^d  - (y+t) z_2^d  - y' z_3^d + (y'+ t) z_4 ^d  $$
The intersection of this with $z_1  - z_2  - z_3 + z_4 =0$ is smooth unless $$ y^{-1/(d-1)} - (y+t)^{-1/(d-1)} - y' ^{-1/(d-1)} + (y'+t)^{1/d-1}=0$$. 
So you get square-root cancellation outside this set for arbitrary $f$. 
A: Estimates for such sums can often be deduced from Deligne's version of the Riemann Hypothesis using computations of the relevant monodromy groups (typically by Katz, using sheaf-theoretic Fourier-transform arguments) and algebraic-group results (what Katz calls the Goursat-Kolchin-Ribet argument).  For analytic number theory purposes, there's a survey called "A study in sums of products" (Fouvry, Kowalski, Michel) that you can find online, with examples and pointers to the literature.
