  In this paper we present some new identities of hypergeometric type for
multiple harmonic sums whose indices are the sequences $(\{1\}^a,c,\{1\}^b),$
$(\{2\}^a,c,\{2\}^b)$ and prove a number of congruences for these sums modulo a
prime $p.$ The congruences obtained allow us to find nice $p$-analogues of
Leshchiner's series for zeta values and to refine a result due to M. Hoffman
and J. Zhao about the set of generators of the multiple harmonic sums of weight
7 and 9 modulo $p$. Moreover, we are also able to provide a new proof of
Zagier's formula for $\zeta^{*}(\{2\}^a,3,\{2\}^b)$ based on a finite identity
for partial sums of the zeta-star series.
