The discharge and plasma plume characteristics of the cylindrical Hall thruster were studied in regimes with external modulations of the applied voltage. It is found that the amplitude and the root mean square (rms) value of the discharge and ion currents increase with the amplitude of the external modulation exhibiting two different regimes. For smaller amplitudes of the modulation voltage, the oscillations amplitude and rms value of the discharge and ion currents follow the amplitude of modulations approximately linearly. For larger voltage modulations, the amplitude and the rms value of the discharge and ion currents grow faster and nonlinearly. In the nonlinear regime, the discharge and the ion currents demonstrate pronounced dependence on the frequency of the external modulations. Moreover, the rms value of the ion current is amplified stronger than the rms value of the discharge current resulting in an increase of the current utilization (of about 5%) and the propellant utilization efficiencies (of about 40%). The thruster efficiency, defined as a product of the current and propellant utilization coefficients, shows an increase of the about 20%. We also present the results of theoretical modeling of a plasma response to driven oscillations in a simplified 1D model of resistive-ionization mode in quasineutral plasma. This modeling demonstrates the nonlinear property of the fundamental breathing mode similar to the experimental results.