  Let $J_{n,m}:=(x_1x_2\cdots x_m,\; x_2x_3\cdots x_{m+1},\; \ldots,\;
x_{n-m+1}\cdots x_n,\; x_{n-m+2}\cdots x_nx_1, \ldots, x_nx_1\cdots x_{m-1})$
be the $m$-path ideal of the cycle graph of length $n$, in the ring
$S=K[x_1,\ldots,x_n]$.
  Let $d=\gcd(n,m)$. We prove that $\operatorname{depth}(S/J_{n,m}^t)\leq d-1$
for all $t\geq n-1$. We show that
$\operatorname{sdepth}(S/J_{n,n-1}^t)=\operatorname{depth}(S/J_{n,n-1}^t)=\max\{n-t-1,0\}$
for all $t\geq 1$. Also, we give some bounds for
$\operatorname{depth}(S/J_{n,m}^t)$ and $\operatorname{sdepth}(S/J_{n,m}^t)$,
where $t\geq 1$.
