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Of the various tumour types, colorectal cancer and brain tumours are still considered among the most serious and deadly diseases in the world. Therefore, many researchers are interested in improving the accuracy and reliability of diagnostic medical machine learning models. In computer-aided diagnosis, self-supervised learning has been proven to be an effective solution when dealing with datasets with insufficient data annotations. However, medical image datasets often suffer from data irregularities, making the recognition task even more challenging. The class decomposition approach has provided a robust solution to such a challenging problem by simplifying the learning of class boundaries of a dataset. In this paper, we propose a robust self-supervised model, called <i>XDecompo</i>, to improve the transferability of features from the pretext task to the downstream task. <i>XDecompo</i> has been designed based on an affinity propagation-based class decomposition to effectively encourage learning of the class boundaries in the downstream task. <i>XDecompo</i> has an explainable component to highlight important pixels that contribute to classification and explain the effect of class decomposition on improving the speciality of extracted features. We also explore the generalisability of <i>XDecompo</i> in handling different medical datasets, such as histopathology for colorectal cancer and brain tumour images. The quantitative results demonstrate the robustness of <i>XDecompo</i> with high accuracy of 96.16% and 94.30% for CRC and brain tumour images, respectively. <i>XDecompo</i> has demonstrated its generalization capability and achieved high classification accuracy (both quantitatively and qualitatively) in different medical image datasets, compared with other models. Moreover, a post hoc explainable method has been used to validate the feature transferability, demonstrating highly accurate feature representations.