Hyperspectral image (HSI) restoration is a challenging research area, covering a variety of inverse problems. Previous works have shown the great success of deep learning in HSI restoration. However, facing the problem of distribution gaps between training HSIs and target HSI, those data-driven methods falter in delivering satisfactory outcomes for the target HSIs. In addition, the degradation process of HSIs is usually disturbed by noise, which is not well taken into account in existing restoration methods. The existence of noise further exacerbates the dissimilarities within the data, rendering it challenging to attain desirable results without an appropriate learning approach. To track these issues, in this article, we propose a supervise-assisted self-supervised deep-learning method to restore noisy degraded HSIs. Initially, we facilitate the restoration network to acquire a generalized prior through supervised learning from extensive training datasets. Then, the self-supervised learning stage is employed and utilizes the specific prior of the target HSI. Particularly, to restore clean HSIs during the self-supervised learning stage from noisy degraded HSIs, we introduce a noise-adaptive loss function that leverages inner statistics of noisy degraded HSIs for restoration. The proposed noise-adaptive loss consists of Stein's unbiased risk estimator (SURE) and total variation (TV) regularizer and fine-tunes the network with the presence of noise. We demonstrate through experiments on different HSI tasks, including denoising, compressive sensing, super-resolution, and inpainting, that our method outperforms state-of-the-art methods on benchmarks under quantitative metrics and visual quality. The code is available at https://github.com/ying-fu/SSDL-HSI.