We ask whether every homologically trivial cyclic action on a symplectic four-manifold extend to a Hamiltonian circle action. By a cyclic action we mean an action of a cyclic group of finite order; it is homologically trivial if it induces the identity map on homology. We assume that the manifold is closed and connected. In the talk, I will give an example of a homologically trivial symplectic cyclic action on a four-manifold that admits Hamiltonian circle actions, and show that is does not extend to a Hamiltonian circle action. I will also discuss symplectic four-manifolds on which every homologically trivial cyclic action extends to a Hamiltonian circle action. I will deduce corollaries on the existence of homologically trivial cyclic actions and on embedding finite-order cyclic subgroups of the group of Hamiltonian symplectomorphisms in circle subgroups. This work applies holomorphic methods to extend combinatorial tools developed for circle actions to study cyclic actions.