We study the propagation dynamics for a bistable delayed reaction-diffusion system with degeneracy in diffusion. This model is used to describe the population suppression of wild mosquitoes via releasing Wolbachia-infected mosquitoes. We first obtain the existence of bistable wave, which is then shown to have a unique speed. Such a uniqueness is then used to completely characterize the sign of this speed in terms of releasing size of mosquitoes. When the speed is positive, we further investigate the long term behavior of solutions starting from a family of ordered and compactly supported initial values; two thresholds for initial values are identified to distinguish the convergence of these solutions to different stable equilibria, and this corresponds to the success or failure of population suppression.