Non-orthogonal multiple access (NOMA)-aided cell-free massive multiple-input multiple-output (CFmMIMO) has been considered as a promising technology to fulfill strict quality of service requirements for ultra-reliable low-latency communications (URLLC). However, finite blocklength coding (FBC) in URLLC makes it challenging to achieve the optimal performance in the NOMA-aided CFmMIMO system. In this paper, we investigate the performance of the NOMA-aided CFmMIMO system with FBC in terms of achievable sum rate (ASR). Firstly, we derive a lower bound (LB) on the ergodic data rate. Then, we formulate an ASR maximization problem by jointly considering power allocation and user equipment (UE) clustering. To tackle such an intractable problem, we decompose it into two sub-problems, i.e., the power allocation problem and the UE clustering problem. A successive convex approximation (SCA) algorithm is proposed to solve the power allocation problem by transforming it into a series of geometric programming problems. Meanwhile, two algorithms based on graph theory are proposed to solve the UE clustering problem by identifying negative loops. Finally, alternative optimization is performed to find the maximum ASR of the NOMA-aided CFmMIMO system with FBC. The simulation results demonstrate that the proposed algorithms significantly outperform the benchmark algorithms in terms of ASR under various scenarios.