The linear constraint of secret key rate capacity is overcome by the twin-field quantum key distribution (QKD). However, the complex phase-locking and phase-tracking technique requirements throttle the real-life applications of the twin-field protocol. The asynchronous measurement-device-independent (AMDI) QKD, also called the mode-pairing QKD, protocol can relax the technical requirements and keep the similar performance of the twin-field protocol. Here, we propose an AMDI-QKD protocol with a nonclassical light source by changing the phase-randomized weak coherent state to a phase-randomized coherent-state superposition in the signal state time window. Simulation results show that our proposed hybrid source protocol significantly enhances the key rate of the AMDI-QKD protocol, while exhibiting robustness to imperfect modulation of nonclassical light sources.