We investigate the dynamical behavior of a delayed HIV infection model with general incidence rate and immune impairment. We derive two threshold parameters, the basic reproduction number R0 and the immune response reproduction number R1. By using Lyapunov functional and LaSalle invariance principle, we prove the global stability of the infection-free equilibrium and the infected equilibrium without immunity. Furthermore, the existence of Hopf bifurcations at the infected equilibrium with CTL response is also studied. By theoretical analysis and numerical simulations, the effect of the immune impairment rate on the stability of the infected equilibrium with CTL response has been studied.