Let M be a right R-module. We call M Rad-H-supplemented iffor each Y M there exists a direct summand D of M such that(Y + D)/D (Rad(M) + D)/D and (Y + D)/Y (Rad(M) + Y )/Y .It is shown that:(1) Let M = M1M2, where M1 is a fully invariant submodule of M.If M is Rad-H-supplemented, thenM1 andM2 are Rad-H-supplemented.(2) Let M = M1 M2 be a duo module and Rad--supplemented. IfM1 is radical M2-sejective (or M2 is radical M1-sejective), then M isRad-H-supplemented. (3) Let M = ni=1Mi be a finite direct sum ofmodules. If Mi is generalized radical Mj-projective for all j > i andeach Mi is Rad-H-supplemented, then M is Rad-H-supplemented.