Let \(p\) be an odd prime number and let \[ S=\sum_{\substack{ \chi\bmod p^3\\ \chi^{p^2}=\chi_0,\;\chi\neq\chi_0}} \sum_{x=1}^{p^3} \left|\sum_{m=1}^h\chi(x+m)\right|^{2r}. \] The author derives the bounds \[ S\ll\begin{cases} p^2h^4+p^5h^2, & r=2,\\ p^2h^6+\min(h,p)^3p^5h+\min(h,p)^2p^6, &r=3.\end{cases} \] These estimates improve previous results due to the author on an average for the non-principal characters \(\chi\) modulo \(p^3\) in the group of order \(p^2\).