Among the promising 6G candidate techniques to support big-data mobile wireless networks, the cell-free massive multi-input-multi-output (CF-M-MIMO) technique has received a great deal of research attentions, where a large number of distributed single-antenna access points (APs), whose ensemble forms a distributed massive MIMO array, simultaneously and jointly serve the single-antenna mobile users. However, CF-M-MIMO results in the low efficiencies for big-data mobile networks when the size of the served area increases. To overcome this challenge, the user-centric approach has been integrated with CF-M-MIMO such that APs only serve a selected subset of mobile users rather than all of them. However, the performance of this approach cannot outperform the traditional CF-M-MIMO in overloaded cases, where the number of mobile users is larger than that of APs. To efficiently implement the big-data aware user-centric cell-free massive MIMO system and enhance the spectral and energy efficiencies while reducing the interference, in this paper we propose an efficient non-orthogonal multiple access (NOMA) and user-centric based CF-M-MIMO scheme over 6G mobile wireless networks. Our proposed scheme develops a more efficient central-processing-unit based bipartite graph matching algorithm to select the optimal mobile users served by each AP. Then, we propose the NOMA-aided power allocation and pilot cluster assignment schemes. The numerical results show that our proposed schemes outperform the existing schemes without applying the NOMA technique in terms of mitigating the interference and enhancing the energy efficiency.