In this paper, we consider the following perturbed second-order Hamiltonian system
−u¨(t)+L(t)u=∇W(t,u(t))+∇G(t,u(t)),∀t∈R,
where W(t,u) is subquadratic near origin with respect to u; the perturbation term G(t,u) is only locally defined near the origin and may not be even in u. By using the variant Rabinowitz's perturbation method, we establish a new criterion for guaranteeing that this perturbed second-order Hamiltonian system has infinitely many homoclinic solutions under broken symmetry situations. Our result improves some related results in the literature