Nonexistence conditions of a solution for the congruence $x_1^k+\cdots+x_s^k\equiv N\pmod{p^n}$

We obtain nonexistence conditions of a solution for of the congruence $x_1^k+\cdots+x_s^k\equiv N\pmod{p^n}$, where $k\geq 2$, $s\geq 2$ and $N$ are integers, and $p^n$ is a prime power. We give nonexistence conditions of the form $(s, N\bmod{p^n})$ for $k=2$, $3$, $4$, $5$, $7$, and of the form $(s, p^n)$ for $k=11$, $13$, $17$, $19$. Furthermore, we complete some tables concerned with Waring's problem in $p$-adic fields that were computed by Hardy and Littlewood.