Spectral computed tomography (CT) can provide attenuation information at different energy levels, which is essential for material decomposition and tissue discrimination. Sparse-view projection can effectively reduce radiation dose but can cause severe artifacts and noise in the reconstructed spectral CT images. Although deep learning reconstruction methods based on convolutional neural networks can improve the image quality, a loss in the tissue detail features is observed. Therefore, a spectral CT reconstruction method based on a hybrid attention module combined with a multiscale feature fusion edge enhancement network (HAMEN) is proposed. The network first extracts edge features of the input images through the edge enhancement module and concatenates them on the images, enriching the input image information. Next, a hybrid attention module is used to generate channel attention and spatial attention maps, which are used to refine the input features. The multiscale feature fusion mechanism is developed at the encoder, and some skip connections are added to minimize feature loss caused by the stacking of convolutional layers. The experimental results show that the peak signal-to-noise ratio of the CT images obtained using the proposed method is 37.64 dB, and the similarity structural index measure is 0.9935. This method can suppress artifacts and noise caused by sparse-view projection while preserving the tissue detail information. Furthermore, the CT image quality is improved for subsequent diagnosis and other works.