  Let $p$ be a prime, let $s \geq 3$ be a natural number and let $A \subseteq
\mathbb{F}_p$ be a non-empty set satisfying $|A| \ll p^{1/2}$. Denoting
$J_s(A)$ to be the number of solutions to the system of equations \[
\sum_{i=1}^{s} (x_i - x_{i+s}) = \sum_{i=1}^{s} (x_i^2 - x_{i+s}^2) = 0, \]
with $x_1, \dots, x_{2s} \in A$, our main result implies that \[ J_s(A) \ll
|A|^{2s - 2 - 1/9}. \] This can be seen as a finite field analogue of the
quadratic Vinogradov mean value theorem. Our techniques involve a variety of
combinatorial geometric estimates, including studying incidences between
cartesian products $A\times A$ and a special family of modular hyperbolae.
