  In this paper, we develop the theory of Hom-Lie algebroids, Hom-Lie
bialgebroids and Hom-Courant algebroids introduced by Cai, Liu and Sheng.
Specifically, we introduce the notions of Hom-Poisson, Hom-Nijenhuis and
Hom-Poisson-Nijenhuis structures on a Hom-Lie algebroid and the notion of
Hom-Dirac structures on a Hom-Courant algebroid. We show that these structures
satisfy similar properties to structures non "Hom-"version. For example, there
exists the hierarchy of a Hom-Poisson-Nijenhuis structure and we have a
relation between Hom-Dirac structures and Maurer-Cartan type equation. Moreover
we show that there exists a one-to-one correspondence between the pairs
consisting of a Poisson structure on $M$ and a Poisson isomorphism for it.
