<p>This paper presents a comprehensive dynamic analysis of a modified Leslie–Gower model with the Beverton-Holt-like function and the additive Allee effect. Using the qualitative and bifurcation theories of dynamical systems, we investigate bifurcations of codimensions 1 through 4, such as Bautin and codimension-4 Bogdanov-Takens bifurcations. Numerical simulations further reveal complex dynamics, including the coexistence of two limit cycles induced by a Bautin bifurcation and three limit cycles induced by a codimension-4 Bogdanov-Takens bifurcation. Through the analysis, the following conclusions are obtained: The strength of the additive Allee effect determines whether and in what form the prey and predator can coexist; A substantial reduction in the Beverton-Holt-like function may weaken the predator’s numerical response to prey density; An increase in alternative food resources raises predator abundance, thereby increasing the extinction risk for the prey. This study is significant for understanding the dynamical behavior of prey affected by the additive Allee effect and generalist predator subject to reproductive competition.</p>