The convolutional neural network (CNN) has made immense progress in the classification of coronavirus disease (COVID-19) X-ray images; however, the convolution structure can only learn the context information of the adjacent spatial positions of the feature map. Hence, to realize a better combination of the global information of chest X-ray images, we propose a network that pays more attention to the interaction of the global and local information by designing the backbone network, ConvNeXt, a convergent attention module, and a long short-term memory network while improving the CNN depth as well. Herein, this experiment classified the images of the COVID-19 Radiography Database dataset, which can be publicly accessed. Compared with the basic model of ConvNeXt, the proposed network displays an improvement in the accuracy, accuracy, and recall by 1.60, 1.23, and 1.76 percentage points, respectively, in the three classification experiments, and it is superior to Vision Transformer and Swin-Transformer in many experimental indicators, with the accuracy, accuracy, recall, and specificity increased to 95.6%, 96.03%, 95.76%, and 97.53%, respectively. Furthermore, the Chest X-ray dataset was selected to further verify the proposed network generalization capability, and the score-CAM algorithm was used to verify its effectiveness. The experimental results show that the proposed network has a high potential for application in the classification of COVID-19 X-ray images.