  In this paper, we introduce the notion of the Hom-Leibniz-Rinehart algebra as
an algebraic analogue of Hom-Leibniz algebroid, and prove that such an
arbitrary split regular Hom-Leibniz-Rinehart algebra $L$ is of the form
$L=U+\sum_\gamma I_\gamma$ with $U$ a subspace of a maximal abelian subalgebra
$H$ and any $I_\gamma$, a well described ideal of $L$, satisfying $[I_\gamma,
I_\delta]= 0$ if $[\gamma]\neq [\delta]$. In the sequel, we develop techniques
of connections of roots and weights for split Hom-Leibniz-Rinehart algebras
respectively. Finally, we study the structures of tight split regular
Hom-Leibniz-Rinehart algebras.
