Let $k=\mathbb{F}_q$ be the finite field with $q=p^r$ elements. Assume that $q$ is odd. Let $\psi$ be a nontrivial additive character of $k$. For $a,b\in k^\times$, consider the Gauss sum $$G(a,b)=\sum_{x\in k^\times}\psi\left(ax+\frac{b}{x}\right).$$ Is there a general formula for $G(a,b)$ in terms of $a,b$?
Any hints, references are appreciated.