In this study, we construct a relaxed second-order time-accurate scheme for the binary phase-field crystal model based on the recent paper [J. Wu, J. Yang and Z. Tan, Eng. Comput. (2023)]. This scheme can significantly improve the consistency between original and modified energies. Using the exponential SAV method, we construct the first- and second-order time-accurate schemes by using the backward Euler formula and the second-order backward difference formula, respectively. Then a new second-order numerical scheme with relaxation is developed. This new numerical scheme satisfies the energy dissipation law. Meanwhile, it is fully decoupled and efficient. Extensive numerical experiments are carried out in 2D and 3D to verify the accuracy and stability of the proposed scheme, as well as its ability to improve the consistency between the original energy and modified energy.