A modifified predator-prey model with a new nonlinear harvesting on predator and gestation delay of prey is studied. It is shown that the stability of interior equilibrium point can switch fifinite times and Hopf bifurcations occur when delay increases through critical values. The properties of the Hopf bifurcations are investigated by the center manifold theorem. Special attention is paid to singularity-induced bifurcations and their state feedback control. Numerical simulations demonstrate the effectiveness of the theoretical results.