For resonant-type absorbers with a back cavity, the improvement of low-frequency sound absorption always depends on increasing the back cavity depth, which hinders its practical application. To realize satisfactory sound absorption performance in low- frequency range (<500Hz) while keeping the depth of the back cavity unchanged, a membrane-cavity-grate (MCG) meta-structure is proposed. The specific physical mechanisms for sound absorption are analyzed from acoustic impedance and energy dissipate perspectives. Furthermore, some key parameters of the MCG unit cell are investigated to optimize the sound absorption performance. On this basis, a meta-structure with 6-unit-cell is proposed. Compared with the structure without the grating, the meta-structure can realize 20% improvement of the average sound absorption coefficient in the range from 300-700Hz with the back cavity depth and the structure size unchanged, which is verified by the high consistency between finite element simulation and acoustic impedance tube experiment results. This work provides a potential avenue for low-frequency noise reduction structure design.