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In the paper, we firstly study the algebraic structures of $\mathbb{Z}_p\mathbb{Z}_{p^k}$-additive cyclic codes and give the generator polynomials andthe minimal spanning set of these codes. Secondly, a necessary and sufficientcondition for a class of $\mathbb{Z}_p\mathbb{Z}_{p^2}$-additive codes whoseGray images are linear (not necessarily cyclic) over $\mathbb{Z}_p$ is given.Moreover, as for some special families of cyclic codes over $\mathbb{Z}_{9}$and $\mathbb{Z}_3 \mathbb{Z}_{9}$, the linearity of the Gray images isdetermined.

Gray Images of Cyclic Codes over $\mathbb{Z}_{p^2}$ and $\mathbb{Z}_p\mathbb{Z}_{p^2}