I am going to assume that by an **additive character** you mean

> an **irreducible representation** $\chi_\alpha : \mathbb{F}^n_q \longrightarrow  \mathbb{C}$, i.e. a group homomorphism from the **additive group** $(\mathbb{F}^n_q ,+)$ to the multiplicative group $(\mathbb{C},*)$

 which we can prove must all take the form 
\begin{equation}\chi_\alpha : \beta \mapsto \exp\left( {\frac{2\pi i \left\langle \alpha ,\beta \right\rangle }{p  }} \right)\end{equation} where $ \left\langle \alpha ,\beta \right\rangle = \sum_i \alpha_i \beta_i $, see [chapter 4 of Tao][1] for a proof of some of these statements and see [Serre][2] or [Fulton & Harris][3] for a general overview of the representation theory perspective on characters. The point is the following 

> If we let \begin{equation} f(x) = \begin{cases} q \psi_x(x) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \\ \end{cases} \end{equation} then the sum you are considering is equal to the Fourier transform of $f$ *i.e.* \begin{equation}  \hat{f}(\alpha) = \frac{1}{q} \sum_{c \in \mathbb{F} _q } f(c) \chi_\alpha(c) =  \sum_{c \in \mathbb{F} _q^* } \psi_c (c) \chi_\alpha(c) \end{equation} see [definition 4.6 in Tao][1].

We apply the Hausdorff-Young inequality [theorem 4.8 in Tao][1] to get that 
\begin{equation}
 \left(\sum_{\alpha \in \mathbb{F} _q }\left| \hat f(\alpha)\right|^{p'} \right)^{\frac{1}{p'}} \leq \left(\sum_{\alpha \in \mathbb{F} _q } |f(\alpha)|^p\right)^{\frac{1}{p}}  = q\left( \sum_{c \in \mathbb{F} _q^* } |\psi_c (c) |^p\right)^{\frac{1}{p}}
\end{equation}
where the LHS is the $l^{q}$-norm, the RHS is the $l^p$-norm, and $p$ satisfies the following  $p^{-1} +q^{-1} = 1 \land 1 \leq p\leq 2$. Plugging in $p = 2$ we get that  

\begin{equation}
 \sum_{\alpha \in \mathbb{F} _q }\left| \hat f(\alpha)\right|^{2} \leq  q\sum_{c \in \mathbb{F} _q^* } |\psi_c (c) |^2
\end{equation}
which is equivalent to saying that 
\begin{equation}
\mathbb{Var}[\hat f] =  \frac{1}{q}\sum_{\alpha \in \mathbb{F} _q }\left| \hat f(\alpha)\right|^{2} \leq  \sum_{c \in \mathbb{F} _q^* } |\psi_c (c) |^2\leq  q-1.
\end{equation}

Finally, if you can prove that at least $n$ many $\alpha$ give the value $| \hat f(a)| = \sqrt b$ then you get that
\begin{equation}
nb +\sup_{\alpha \in \mathbb{F} _q }\left| \hat f(a)\right|^2 =  \sum_{\alpha \in S}\left| \hat f(\alpha)\right|^{2}  + \sup_{\alpha \in \mathbb{F} _q }\left| \hat f(a)\right|^2 \leq  \sum_{\alpha \in \mathbb{F} _q }\left| \hat f(\alpha)\right|^{2} \leq q-1
\end{equation}

which gives you that the maximum value is at most 



\begin{equation}
 \sup_{\alpha \in \mathbb{F} _q }\left|\sum_{c \in \mathbb{F} _q^* } \psi_c (c) \chi_\alpha(c) \right| = \sup_{\alpha \in \mathbb{F} _q }\left| \hat f(a)\right| \leq \sqrt{q-1-nb}
\end{equation}



  [1]: https://www.google.com/books/edition/Additive_Combinatorics/xpimQMtn5-IC?hl=en&gbpv=1&printsec=frontcover
  [2]: https://www.google.com/books/edition/Linear_Representations_of_Finite_Groups/9mT1BwAAQBAJ?hl=en&gbpv=1&printsec=frontcover
  [3]: https://www.google.com/books/edition/Representation_Theory/6TwmBQAAQBAJ?hl=en&gbpv=1&printsec=frontcover