The COVID-19 pandemic, with its multiple variants, has placed immense pressure on the global healthcare system. An early effective screening and grading become imperative towards optimizing the limited available resources of the medical facilities. Computed tomography (CT) provides a significant non-invasive screening mechanism for COVID-19 infection. An automated segmentation of the infected volumes in lung CT is expected to significantly aid in the diagnosis and care of patients. However, an accurate demarcation of lesions remains problematic due to their irregular structure and location(s) within the lung. A novel deep learning architecture, Mixed Attention Deeply Supervised Network (MiADS-Net), is proposed for delineating the infected regions of the lung from CT images. Incorporating dilated convolutions with varying dilation rates, into a mixed attention framework, allows capture of multi-scale features towards improved segmentation of lesions having different sizes and textures. Mixed attention helps prioritise relevant feature maps to be probed, along with those regions containing crucial information within these maps. Deep supervision facilitates discovery of robust and discriminatory characteristics in the hidden layers at shallower levels, while overcoming the vanishing gradient. This is followed by estimating the severity of the disease, based on the ratio of the area of infected region in each lung with respect to its entire volume. Experimental results, on three publicly available datasets, indicate that the MiADS-Net outperforms several state-of-the-art architectures in the COVID-19 lesion segmentation task; particularly in defining structures involving complex geometries.