Objective. The risk of radiobiological stochastic effects associated with neutrons is strongly energy dependent. Recent Monte Carlo studies simulating neutron-irradiated nuclear DNA have demonstrated that this energy dependence is correlated with the relative biological effectiveness (RBE) of neutrons to inflict DNA damage clusters that contain difficult-to-repair double-strand breaks. However, these previous investigations were either limited to modeling direct radiation action or considered the effects of both direct and indirect action together without distinguishing between the two. In this study, we aimed to quantify the influence of indirect action in neutron irradiation scenarios and acquire novel estimations of the energy-dependent neutron RBE for inducing DNA damage clusters due to both direct and indirect action. Approach. We explored the role of indirect action in neutron-induced DNA damage by integrating a validated indirect action model into our existing simulation pipeline. Using this pipeline, we performed track-structure simulations of monoenergetic neutron irradiations (1 eV to 10 MeV) in a nuclear DNA model and analyzed the resulting simple and clustered DNA lesions. We repeated the irradiation simulations for 250 keV x-rays that acted as our reference radiation. Main results. Including indirect action significantly increased the occurrence of DNA lesions. We found that indirect action tends to amplify the damage due to direct action by inducing DNA lesions in the vicinity of directly-induced lesions, resulting in additional and larger damage clusters. Our neutron RBE results are qualitatively similar to but lower in magnitude than the established radiation protection factors and the results of previous similar investigations, due to the greater relative impact of indirect action in photon-induced damage than in neutron-induced damage. Significance. Although our model for neutron-induced DNA damage has some important limitations, our findings suggest that the energy-dependent risk of neutron-induced stochastic effects may not be completely modeled alone by the relative potential of neutrons to inflict clustered lesions via direct and indirect action in DNA damage.