  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 and
the minimal spanning set of these codes. Secondly, a necessary and sufficient
condition for a class of $\mathbb{Z}_p\mathbb{Z}_{p^2}$-additive codes whose
Gray 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 is
determined.
