  In recent years, the congruence $$ \sum_{\substack{i+j+k=p\\ i,j,k>0}}
\frac1{ijk} \equiv -2 B_{p-3} \pmod{p}, $$ first discovered by the last author
have been generalized by either increasing the number of indices and
considering the corresponding supercongruences, or by considering the
alternating version of multiple harmonic sums. In this paper, we prove a family
of similar supercongruences modulo prime powers $p^r$ with the indexes summing
up to $mp^r$ where $m$ is coprime to $p$, where all the indexes are also
coprime to $p$.
