Lower estimates of sums of polynomial characters

An infinite sequence of primes $p$ is formulated, and for each $p$ polynomials of form $ax^n+b$, $(a,p=(b,p)=1$, are indicated such that
$$\sum_{x=1}^p\biggl(\frac{ax^n+b}p\biggr),\quad n\asymp\frac p{\log p}.$$