  The harmonic number $H_k=\sum_{j=1}^k1/j(k=1,2,3\cdots)$ play an important
role in mathematics. Let $p>3$ be a prime. In this paper, we establish a number
of congruences with the form $\sum_{k=1}^{p-1}k^mH_k^n(\mod p^2)$ for
$m=1,2\cdots,p-2$ and $n=1,2,3$.
