The brightness of betatron radiation generated by laser wakefield accelerated electrons can be increased by utilizing the laser driver with shorter duration at the same energy. Such shortening is possible by pulse compression after its nonlinear self-phase modulation in a thin plate. However, this method can lead to a rather complex femtosecond-scale time structure of the pulse. In this work, the results of numerical simulations show that the presence of prepulses containing a few percent of the main pulse energy can significantly alter the acceleration process and lead to either lower or higher energies of accelerated electrons and generated photons, depending on the prepulse parameters. Simultaneously, the presence of a pedestal inhibits the acceleration process lowering the brightness of the betatron source. Furthermore, postpulses following the main pulse are not found to have a significant effect on betatron radiation.