An estimate for incomplete mixed character sums and applications

Let $q$ be a prime power and $m>1$ be any integer. Let $\mathbb F_{q^m}$ be the finite field of order $q^m$ and $\theta\in\mathbb F_{q^m}$ be such that $\mathbb F_{q^m} = \mathbb F(\theta)$. We obtain a nontrivial bound for the mixed character sum $\sum_{x \in\mathbb F}\chi(\theta+x)\psi(x)$, where $\chi$ and $\psi$ are multiplicative and additive characters of $\mathbb F_{q^m}$ and $\mathbb F$, respectively, using function field methods. As an application of our main result, we prove that for fixed $m$ and sufficiently large prime powers $q$, that satisfy certain conditions, $\mathbb F_{q^m}/\mathbb F$ possesses the weak line property for primitive normal elements. In particular, our result is a strengthening of existing results.