State-of-the-art free-space continuous-variable quantum key distribution (CV-QKD) protocols use phase reference pulses to modulate the wavefront of a real local oscillator at the receiver, thereby compensating for wavefront distortions caused by atmospheric turbulence. It is normally assumed that the wavefront distortion in these phase reference pulses is identical to the wavefront distortion in the quantum signals, which are multiplexed during transmission. However, in many real-world deployments, there can exist a relative wavefront error (WFE) between the reference pulses and quantum signals, which, among other deleterious effects, can severely limit secure key transfer in satellite-to-Earth CV-QKD. In this work, we introduce novel machine learning-based wavefront correction algorithms, which utilize multi-plane light conversion for decomposition of the reference pulses and quantum signals into the Hermite-Gaussian (HG) basis, then estimate the difference in HG mode phase measurements, effectively eliminating this problem. Through detailed simulations of the Earth-satellite channel, we demonstrate that our new algorithm can rapidly identify and compensate for any relative WFEs that may exist, whilst causing no harm when WFEs are similar across both the reference pulses and quantum signals. We quantify the gains available in our algorithm in terms of the CV-QKD secure key rate. We show channels where positive secure key rates are obtained using our algorithms, while information loss without wavefront correction would result in null key rates.