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Convolutional Neural Networks have demonstrated dermatologist-level performance in the classification of melanoma from skin lesion images, but prediction irregularities due to biases seen within the training data are an issue that should be addressed before widespread deployment is possible. In this work, we robustly remove bias and spurious variation from an automated melanoma classification pipeline using two leading bias unlearning techniques. We show that the biases introduced by surgical markings and rulers presented in previous studies can be reasonably mitigated using these bias removal methods. We also demonstrate the generalisation benefits of unlearning spurious variation relating to the imaging instrument used to capture lesion images. Our experimental results provide evidence that the effects of each of the aforementioned biases are notably reduced, with different debiasing techniques excelling at different tasks.