On a kind of generalized multi-harmonic sums

Let $p$ be an odd prime, Jianqiang Zhao has established a curious congruence, which is $$\sum_{i+j+k=p \atop i,j,k > 0} \frac{1}{ijk} \equiv -2B_{p-3}\pmod p ,$$ where $B_{n}$ denotes the $n$-th Bernoulli number. In this paper, we will generalize this kind of sums and prove a family of similar congruences modulo prime powers $p^r$.