The study and control of resonant instabilities in magnetised plasmas is of fundamental interest over a wide range of applications from industrially relevant plasmas to plasma sources for spacecraft propulsion. In this work electrostatic probes were employed to measure a 4-20 kHz instability in the ion saturation current downstream of an electric double layer (DL) in an expanding helicon plasma source. The amplitude and frequency of the instability were found to vary in inverse proportion to the operating argon gas pressure (0.2-0.6 mTorr) and in direct proportion to the applied rf power (100-600 W) and applied solenoid current (3-8 A). A spatially resolved characterisation of the maximum instability amplitude determined two radial maxima, corresponding to the locations of most positive radial ion density gradient. Control and inhibition of the instability were achieved through the application of a kHz voltage amplitude modulation to the 13.56 MHz radio-frequency (rf) power supplied to the helicon antenna. Through the application of voltage amplitude modulations in the frequency range 2-12 kHz the instability was reduced by up to 65 %, exhibiting a greater reduction at higher applied modulation frequencies. This effect is described through a variation in the radial ion density gradient via asymmetrically attenuated ion acoustic density perturbations induced by the applied voltage modulation. The application of voltage amplitude modulations has been demonstrated as a potential control mechanism for density gradient driven instabilities in magnetised plasmas.