Let $p$ be a prime, and $\mathbb{F}_p$ be a finite field with $p$ elements. let $\chi$ be a multiplicative character from $\mathbb{F}_p^{\times}$ to $\mathbb{C}$, where $\mathbb{F}_p^{\times}=\{x\in\mathbb{F}_p: x\ne 0\}$. Recently, I met the following sum: $$\sum_{x=0}^{p-1}\chi(a^x-1),$$ where $a>1$ is a positive integer with $p\nmid a$.

Wonder if anybody knows some references concerning the sums of this type.

Thank you for your supposr

Let $p$ be a prime number and $\mathbb{F}_p$ be a finite field with $p$ elements. Let $\chi$ be a multiplicative character from $\mathbb{F}_p^{\times}$ to $\mathbb{C}$, where $\mathbb{F}_p^{\times}=\{x\in\mathbb{F}_p: x\ne 0\}$. 

Recently, I met the following sum: $$\sum_{x=0}^{p-1}\chi(a^x-1),$$ where $a>1$ is a positive integer with $p\nmid a$.

Question. Does anybody know some references concerning sums of this form?