A curious congruence modulo prime powers

Zhao established a curious congruence, i.e., for any prime p≥3p≥3,∑i+j+k=pi,j,k>01ijk≡−2Bp−3(modp). In this note we prove that for any prime p≥3p≥3 and positive integer r,∑i+j+k=pri,j,k∈Pp1ijk≡−2pr−1Bp−3(modpr), where PnPn denotes the set of positive integers which are prime to n.