  In the present paper, we determine the sums
$\sum_{j=1}^{p-1}\frac{H_j^{(s_1)}H_j^{(s_3)}}{j^{s_2}}$ and
$\sum_{j=1}^{p-1}\frac{H_j^{(s_1)}H_j^{(s_3)}H_j^{(s_4)}}{j^{s_2}}$ modulo $p$
and modulo $p^2$ in certain cases. This is done by using multiple harmonic sums
of length three and four, as well as, many other results. In addition, We
recover three congruences conjectured by Z.-W Sun and solved later by the
author himself and R. Me\v{s}trovi\'c.
