On cubic exponential sums and Gauss sums

Let $e_q$ be a nontrivial additive character of a finite field $\mathbb F_q$ of order $q\equiv1\pmod3$, and let $\psi$ be a cubic multiplicative character of $\mathbb F_q$, $\psi(0)=0$. Consider the cubic Gauss sum and the cubic exponential sum
\begin{equation*} G(\psi)=\sum_{z\in\mathbb F_q}e_q(z)\psi(z),\quad C(w)=\sum_{z\in\mathbb F_q}e_q\Bigl(\frac{z^3}w-3z\Bigr),\quad w\in\mathbb F_q\quad w\neq0. \end{equation*}
For all nonzero $a,b\in\mathbb F_q$, $ab\neq0$, it is proved that
\begin{equation*} \frac1q\sum_nC(an)C(bn)\psi(n)+\frac1q\psi(ab)G(\psi)^2=\bar\psi(ab)\psi(a-b)\overline{G(\psi)}, \end{equation*}
where summation runs over all nonzero $n\in\mathbb F_q$.