$H$-supplemented modules with respect to a preradical

Let $M$ be a right $R$-module and $\tau$ a preradical. We call $M$ $\tau$-$H$-supplemented if for every submodule $A$ of $M$ there exists a direct summand $D$ of $M$ such that $(A + D)/D \subseteq \tau(M/D)$ and $(A + D)/A \subseteq \tau(M/A)$. Let $\tau$ be a cohereditary preradical. Firstly, for a duo module $M = M_{1} \oplus M_{2}$ we prove that $M$ is $\tau$-$H$-supplemented if and only if $M_{1}$ and $M_{2}$ are $\tau$-$H$-supplemented. Secondly, let $M=\oplus_{i=1}^nM_i$ be a $\tau$-supplemented module. Assume that $M_i$ is $\tau$-$M_j$-projective for all $j > i$. If each $M_i$ is $\tau$-$H$-supplemented, then $M$ is $\tau$-$H$-supplemented. We also investigate the relations between $\tau$-$H$-supplemented modules and $\tau$-($\oplus$-)supplemented modules.