  We construct homology with trivial coefficients of Hom-Leibniz $n$-algebras.
We introduce and characterize universal ($\alpha$)-central extensions of
Hom-Leibniz $n$-algebras. In particular, we show their interplay with the
zeroth and first homology with trivial coefficients. When $n=2$ we recover the
corresponding results on universal central extensions of Hom-Leibniz algebras.
The notion of non-abelian tensor product of Hom-Leibniz $n$-algebras is
introduced and we establish its relationship with the universal central
extensions. We develop a generalization of the concept and properties of
unicentral Leibniz algebras to the setting of Hom-Leibniz $n$-algebras.
