Hypergraphs provide a powerful framework for modeling polyadic relationships, generalizing classical graph structures. $t$-Cayley hypergraphs, a natural extension of Cayley graphs, are inherently vertex-transitive under the automorphism group ( $\mathrm{A}\mathrm{u}\mathrm{t}$). However, this vertex-transitivity is not necessarily preserved when replacing $\mathrm{A}\mathrm{u}\mathrm{t}$ with non-injective endomorphisms ( ${\mathrm{E}\mathrm{n}\mathrm{d}}^{\prime }$). This work investigates the precise conditions under which $t$-Cayley hypergraphs of finite cyclic groups retain this property. Our main result establishes that a $t$-Cayley hypergraph $H$ over the cyclic group ${\mathbb{Z}}_n$ is ${\mathrm{E}\mathrm{n}\mathrm{d}}^{\prime }$-vertex-transitive if $t$ divides $n$.