In this article, prespecified time synchronization (PTS) is addressed for dynamic networks with impulsive effects and directed topology. The less conservative cases considering the synchronizing and desynchronizing impulses simultaneously are presented, and the discontinuous nonlinear functions are also involved. Moreover, by constructing a quantized prespecified time controller, some constraints on an impulsive interval are relaxed, which makes our results more widely applicable. Sufficient conditions are derived to realize the PTS of impulsive dynamic networks. In addition, the synchronization time of a PTS is independent of any parameters. Numerical examples are provided to show the validity of our theoretical results.