<p>The weak Allee effect on prey in predator–prey systems has been extensively explored in numerous studies. However, the comprehensive dynamics research on its interaction with Michaelis–Menten type predator harvesting remains to be improved. This study focuses on this issue and investigates the impacts of these two factors on the proposed predator–prey system. Firstly, we prove the positivity and boundedness of solutions, calculate the boundary and positive equilibria, and derive their local stability conditions. Subsequently, by applying the qualitative and bifurcation theories of dynamical systems, we study codimension 1 bifurcations such as saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation, as well as Bautin bifurcation and Bogdanov–Takens bifurcations of codimension 2 and 3. The research results reveal the crucial role of the weak Allee effect in the system. The combination of the weak Allee effect and Michaelis–Menten type harvesting causes the system to exhibit multistable behavior. Finally, several numerical simulations are given to verify the theoretical results, and biological interpretations of bifurcations and phase diagrams are provided. This study offers some new perspectives for understanding the complex dynamics of predator–prey systems, contributing to a comprehensive analysis of the stability of ecological systems and the interaction mechanisms among species.</p>