  In this paper, we introduce
$\mathbb{Z}_{p^r}\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$-additive cyclic codes for
$r\leq s$. These codes can be identified as $\mathbb{Z}_{p^s}[x]$-submodules of
$\mathbb{Z}_{p^r}[x]/\langle x^{\alpha}-1\rangle \times
\mathbb{Z}_{p^r}[x]/\langle x^{\beta}-1\rangle\times
\mathbb{Z}_{p^s}[x]/\langle x^{\gamma}-1\rangle$. We determine the generator
polynomials and minimal generating sets for this family of codes. Some previous
works has been done for the case $p=2$ with $r=s=1$, $r=s=2$, and $r=1,s=2$.
However, we show that in these previous works the classification of these codes
were incomplete and the statements in this paper complete such classification.
We also discuss the structure of separable
$\mathbb{Z}_{p^r}\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$-additive cyclic codes and
determine their generator polynomials. Further, we also study the duality of
$\mathbb{Z}_{p^s}[x]$-submodules. As applications, we present some examples and
construct some optimal binary codes.
