A five-leg topology is proposed for a single-phase fault in the six-leg structure of a modular multilevel unified power quality conditioner (MMC-UPQC). This new topology enables power quality compensation in fault conditions. First, after an analysis of the mathematical model of MMC-UPQC in the series and parallel sides, a hybrid model predictive control (H-MPC) is proposed for those sides of the MMC-UPQC. The proposed control method combines finite control set model predictive control (FCS-MPC) and fast model predictive control (Fast-MPC). Then, the computational burden of the control method is reduced by constructing independent cost functions on both sides and the decoupled control of the five-leg topology is also achieved. Finally, compared to traditional linear (for example PI) and nonlinear control strategies (for example passivity-based control, PBC), the proposed hybrid model predictive control has certain advantages in voltage compensation, negative sequence voltage suppression, and harmonic current compensation. It can also avoid complex parameter tuning and coordinate transformation to some extent. Simulation results demonstrate the feasibility and superiority of the proposed control method.