  Let $\Gamma$ be an undirected and simple graph. A set $ S $ of vertices in
$\Gamma$ is called a {cyclic vertex cutset} of $\Gamma$ if $\Gamma - S$ is
disconnected and has at least two components containing cycles. If $\Gamma$ has
a cyclic vertex cutset, then it is said to be {cyclically separable}. The
{cyclic vertex connectivity} of $\Gamma$ is the minimum of cardinalities of the
cyclic vertex cutsets of $\Gamma$. The {power graph} $\mathcal{P}(G)$ of a
group $G$ is the undirected and simple graph whose vertices are the elements
$G$ and two vertices are adjacent if one of them is the power of other in $G$.
In this paper, we first characterize the finite $ p $-groups ($p$ is a prime
number) whose power graphs are cyclically separable in terms of their maximal
cyclic subgroups. Then we characterize the finite $ p $-groups whose power
graphs have equal vertex connectivity and cyclic vertex connectivity.
