At present, most hyperspectral (HS) sharpening methods have not fully utilized the feature correlation between adjacent bands in HS images, nor have they explored the problem of feature uncertainty generated by the model during the fusion process. This may lead to inaccurate fusion features generated by the model, resulting in spatial and spectral distortions in the fusion results. To address these issues, we propose an uncertainty-guided memory network (UMNet) for HS pansharpening. A spatial-spectral recurrent fusion unit (SRFU) is designed based on the concept of temporal data modeling, which utilizes the correlation between adjacent bands to fuse spectral and spatial features from PAN and LRHS images. In SRFU, a state memory interaction unit (SMIU) is constructed based on non-negative matrix factorization (NMF) to learn the global spatial-spectral dependency of PAN and HS images in the recurrent state space. Moreover, based on uncertainty theory, we define two spatial-spectral uncertainty-guided loss functions for the HS pansharpening task to train the model step by step, ensuring that the network can reconstruct more accurate spectral and spatial features. Extensive experiments on three widely used datasets demonstrate that, compared with some state-of-the-art (SOTA) methods, the proposed UMNet has achieved significant improvements in both spatial and spectral quality metrics. The source code of this work will be released on GitHub.