Let $\mathbb{Z}_{p}$ be the ring of residue classes modulo a prime $p$. The $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic codes of length $(α,β)$ is identify as $\mathbb{Z}_{p}[u,v][x]$-submodule of $\mathbb{Z}_{p}[x]/\langle x^α-1\rangle \times \mathbb{Z}_{p}[u,v][x]/\langle x^β-1\rangle$ where $\mathbb{Z}_{p}[u,v]=\mathbb{Z}_{p}+u\mathbb{Z}_{p}+v\mathbb{Z}_{p}$ with $u^{2}=v^{2}=uv=vu=0$. In this article, we obtain the complete sets of generator polynomials, minimal generating sets for cyclic codes with length $β$ over $\mathbb{Z}_{p}[u,v]$ and $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic codes with length $(α,β)$ respectively. We show that the Gray image of $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic code with length $(α,β)$ is either a QC code of length $4α$ with index $4$ or a generalized QC code of length $(α,3β)$ over $\mathbb{Z}_{p}$. Moreover, some structural properties like generating polynomials, minimal generating sets of $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive constacyclic code with length $(α,p-1)$ are determined.