Let M be a right R-module and τ a preradical. We call M τ-H-supplemented if forevery submodule A of M there exists a direct summand D of M such that (A+D)/D ⊆ τ(M/D) and (A+D)/A ⊆ τ(M/A). Let τ be a cohereditary preradical. Firstly, for a duo module M = M1 ⊕M2 we prove that M is τ-H-supplemented if and only if M1 and M2 are τ-H-supplemented. Secondly, let M = ⊕n i=1Mi be a τ-supplemented module. Assume that Mi is τ-Mj-projective for all j > i. If each Mi is τ-H-supplemented, then M is τ-H-supplemented. We also investigate the relations between τ-H-supplemented modules and τ-(⊕-)supplemented modules.
