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

Hom-Poisson-Nijenhuis structures on Hom-Lie algebroids and Hom-Dirac structures on Hom-Courant algebroids