In this paper, we first introduce the notion of twisted O-operators on a Hom-Lie-Yamaguti algebra by a given (2, 3)-cocycle with coefficients in a representation. We show that a twisted O-operator induces a Hom-Lie-Yamaguti structure. We also introduce the notion of a weighted Reynolds operator on a Hom-Lie-Yamaguti algebra, which can serve as a special case of twisted O-operators on Hom-Lie-Yamaguti algebras. Then, we define a cohomology of twisted O-operator on Hom-Lie-Yamaguiti algebras with coefficients in a representation. Furthermore, we introduce and study the Hom-NS-Lie-Yamaguti algebras as the underlying structure of the twisted O-operator on Hom-Lie-Yamaguti algebras. Finally, we investigate the twisted O-operator on Hom-Lie-Yamaguti algebras induced by the twisted O-operator on Hom-Lie algebras.