Self-supervised speech representation learning aims to extract meaningful factors from the speech signal that can later be used across different downstream tasks, such as speech and/or emotion recognition. Existing models, such as HuBERT, however, can be fairly large thus may not be suitable for edge speech applications. Moreover, realistic applications typically involve speech corrupted by noise and room reverberation, hence models need to provide representations that are robust to such environmental factors. In this study, we build on the so-called DistilHuBERT model, which distils HuBERT to a fraction of its original size, with three modifications, namely: (i) augment the training data with noise and reverberation, while the student model needs to distill the clean representations from the teacher model; (ii) introduce a curriculum learning approach where increasing levels of noise are introduced as the model trains, thus helping with convergence and with the creation of more robust representations; and (iii) introduce a multi-task learning approach where the model also reconstructs the clean waveform jointly with the distillation task, thus also acting as an enhancement step to ensure additional environment robustness to the representation. Experiments on three SUPERB tasks show the advantages of the proposed method not only relative to the original DistilHuBERT, but also to the original HuBERT, thus showing the advantages of the proposed method for ``in the wild'' edge speech applications.