In this paper, we study $\mathbb{Z}_p\mathbb{Z}_p[u]$-additive codes, where $p$ is prime and $u^{2}=0$. In particular, we determine a Gray map from $ \mathbb{Z}_p\mathbb{Z}_p[u]$ to $\mathbb{Z}_p^{ α+2 β}$ and study generator and parity check matrices for these codes. We prove that a Gray map $Φ$ is a distance preserving map from ($\mathbb{Z}_p\mathbb{Z}_p[u]$,Gray distance) to ($\mathbb{Z}_p^{α+2β}$,Hamming distance), it is a weight preserving map as well. Furthermore we study the structure of $\mathbb{Z}_p\mathbb{Z}_p[u]$-additive cyclic codes.