This paper is devoted to the construction of Hom-Leibniz H-pseudoalgebras, which unify Hom-Lie H-pseudoalgebras, Leibniz H-pseudoalgebras and Hom-Leibniz algebras. Firstly, we give the construction theorem and obtain a class of Hom-Leibniz H-pseudoalgebras. We also construct Hom-Leibniz H-pseudoalgebras from different perspectives, including Leibniz H-pseudoalgebras, Hom-Leibniz (resp. Hom-Lie, Hom-associative) H-pseudoalgebras and their representations, Hom-Leibniz (resp. Hom-associative) algebras. Then we give some properties of the representations of Hom-Leibniz H-pseudoalgebras. Finally, the annihilation algebras of Hom-Leibniz H-pseudoalgebras are investigated.