In recent years, self-supervised learning (SSL) has achieved tremendous success in various speech tasks due to its power to extract representations from massive unlabeled data. However, compared with tasks such as speech recognition (ASR), the improvements from SSL representation in speech separation (SS) and enhancement (SE) are considerably smaller. Based on HuBERT, this work investigates improving the SSL model for SS and SE. We first update HuBERT's masked speech prediction (MSP) objective by integrating the separation and denoising terms, resulting in a multiple pseudo label pre-training scheme, which significantly improves HuBERT's performance on SS and SE but degrades the performance on ASR. To maintain its performance gain on ASR, we further propose an adapter-based architecture for HuBERT's Transformer encoder, where only a few parameters of each layer are adjusted to the multiple pseudo label MSP while other parameters remain frozen as default HuBERT. Experimental results show that our proposed adapter-based multiple pseudo label HuBERT yield consistent and significant performance improvements on SE, SS, and ASR tasks, with a faster pre-training speed, at only marginal parameters increase.