Unsteady electrostatic forcing is investigated as a method for manipulating turbulent plasma behaviour within Hall effect thrusters and similar cross-field plasma devices using a simplified 1D-3V azimuthal electrostatic particle-in-cell simulation. A wide range of axial electric field forcing frequencies from 1 MHz up to 10 GHz at amplitudes of 10 V/cm, 50 V/cm and 100 V/cm are applied to the plasma and the response is evaluated against a baseline case defined by community benchmark LANDMARK Test Case 1. 'Tailoring' of plasma parameters such as the electron cross-field mobility is demonstrated via manipulation of the electron drift instability using unsteady forcing. Excitation of the unstable electron cyclotron modes of the electron drift instability is shown to be able to produce a reduction of the resultant electron cross-field mobility of the plasma by up to 50% compared to the baseline value. Additionally, forcing at the electron cyclotron frequency appears to be capable of increasing cross-field mobility by up to 2000%. Implications of the results for direct drive electric propulsion systems and improved current utilisation efficiencies for Hall effect thrusters are discussed.