  In the category of Hom-Leibniz algebras we introduce the notion of
representation as adequate coefficients to construct the chain complex to
compute the Leibniz homology of Hom-Leibniz algebras. We study universal
central extensions of Hom-Leibinz algebras and generalize some classical
results, nevertheless it is necessary to introduce new notions of
$\alpha$-central extension, universal $\alpha$-central extension and
$\alpha$-perfect Hom-Leibniz algebra. We prove that an $\alpha$-perfect Hom-Lie
algebra admits a universal $\alpha$-central extension in the categories of
Hom-Lie and Hom-Leibniz algebras and we obtain the relationships between both.
In case $\alpha = Id$ we recover the corresponding results on universal central
extensions of Leibniz algebras.

Rings and Algebras