Theorem 2 Let $k$ be an integer $\geqslant 3$ and $l$ be a positive integer. Let
$$f(x)=a_{k} x^{k}+a_{k-1} x^{k-1}+\cdots+a_{1} x+a_{0}$$

where $a_{i}(i=0,1,2, \cdots, k)$ are all integers. When $p$ is a prime and $\left(a_{k}, a_{k-1}, \cdots, \dot{a}_{1}, p\right)=1$, we have
$$\left|\sum_{x=1}^{p^{t}} e^{2 x i \frac{t(x)}{p^{l}}}\right| \leqslant C_{1}(k) p^{l\left(i-\frac{1}{k}\right)}$$

where
$$C_{1}(k)=\left\{\begin{aligned} 1, & \text { when } p \geqslant(k-1)^{\frac{2 k}{k-2}} \text {; } \\ k^{2 / k}, & \text { when }(k-1)^{\frac{2 k}{k-2}}>p \geqslant(k-1)^{\frac{k}{k-2}} \text {; } \\ k^{3 / k}, & \text { when }(k-1)^{\frac{k}{k-2}}>p>k \text {; } \\ (k-1) & k^{3 / k}, \text { when } p \leqslant k \text {; } \end{aligned}\right.$$