Let $D$ be a relation on a finite set such that $D \subseteq D^2$. If $ (x,y) \in D$ where $x$ is not on a cycle in $\Gamma(D)$ the associated digraph of $D$ then there is a $z$ such that $(x,z),(z,z),(z,y) \in D$.
Certainly there is a $z$ such that $(x,z),(z,y) \in D$. It is also true that there must be a directed path in $\Gamma(D)$ of every positive integer length from $x$ to $y$.
I cannot find a counter example nor come up with a proof. Note that I am not assuming that $x,y,z$ are necessarily distinct.
Take the finite set to be $\{x, z_1, z_2, z_3, y\}$. Let $D$ be the relations $\{(x, y), (x, z_i), (z_i, z_j), (z_j, y): 1 \leq i \neq j \leq 3\}$. Then $D\subset D^2$, and $x$ does not lie in any cycle(since it only has out edges). However, there is even no $z$ with $(z, z) \in D$.