Tridimensional character sums with polynomial arguments and applications

Let $p$ be a large prime and $χ$ a non-trivial Dirichlet character modulo $p$. We study the character sum \[ \sum_{a \sim A} \sum_{b \sim B} \sum_{c \sim C} α(a,b) β(c)χ(f(a) + bc), \] where $z \sim Z$ means $Z\le z < 2Z$, $f \in \mathbb{Z}[X]$ is of small degree $k$, and $\boldsymbolα=(α(a,b))_{a\s