Q.Should I prune, or dig and replace, this Indian Hawthorne Bush (riddled with leaf spots)?
If not an Indian Hawthorne bush, what is its name and how to provide care?

Certified GKH Gardening Expert
To prevent reinfection, rake up all the leaves that have fallen and dispose of them in the trash. Then you can use a fungicide or neem oil to treat the fungus. If that doesn't take care of it, you may consider replacing the shrub.
https://www.gardeningknowhow.com/plant-problems/disease/black-spot-fungus.htm
https://www.gardeningknowhow.com/ornamental/shrubs/indian-hawthorn/indian-hawthorn-pruning.htm
https://www.gardeningknowhow.com/ornamental/shrubs/indian-hawthorn/growing-indian-hawthorn.htm