How can COVID patients and patients with respiratory problems benefit from breathing exercises?

COVID-19 attacks the lungs and respiratory system, sometimes resulting in significant damage. Even in milder cases and even after symptoms like coughing and fever go away, some patients report lung and breathing problems. Recovering lung function may require therapy and exercise for months.
Johns Hopkins Hospital recommends deep breathing exercises to help restore diaphragm function and increase lung capacity. Deep breathing exercises may also help with feelings of distress and anxiety of going through a serious illness. 
If you’re recovering from COVID-19 or another lung condition, you may try deep diaphragmatic breathing for 1 minute while lying on your back, while lying on your stomach, while sitting down and while standing. Consider these as 4 separate phases and only move onto the next phase once you can complete the previous one without feeling out of breath.  
Here’s another exercise that can help patients with respiratory symptoms by maximizing the amount of oxygen that reaches the lungs: 
Breathe in deeply for 5 times (through the mouth, if you can), each time holding your breath for 5 seconds. For the sixth time, breathe in deeply and exhale with a powerful cough.
After doing two cycles, rest for a couple of minutes. Then, lie down flat on your stomach and slowly breathe in and out through the nose for 10 minutes. Lying on your stomach is essential since a majority of lung tissue is on your back and lying prone helps to increase the amount of oxygen reaching the lungs.  
Breathing exercises, including this one, is not meant to treat COVID or any other disease.

You can find more on diaphragmatic breathing here.