  The notions of transposed Hom-Poisson and Hom-pre-Lie Poisson algebras are
introduced. Their bimodules and matched pairs are defined and the relevant
properties and theorems are given. The notion of Manin triple of transposed
Hom-Poisson algebras is introduced, and its equivalence to the transposed
Hom-Poisson bialgebras is investigated. The notion of $\mathcal{O}$-operator is
exploited to illustrate the relations existing between transposed Hom-Poisson
and Hom-pre-Lie Poisson algebras.
