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Hello. Thanks for the question.
It sounds like you are most likely dealing with asthma. This is the most common cause of coughing in cats. Asthma in cats is similar to asthma in people. They can breathe in, but the airways spasm as they try to exhale. This leads to wheezing and coughing. When it gets bad enough, we can have the right middle lung lobe collapse down. This is the smallest lung lobe, so that's why it's the one that we typically see collapsed.
There is no specific therapy for a collapsed lobe. We let it re-inflate on it's own. In some cats, it does not re-inflate. The other lung lobes will actually get larger to make up for the loss of that right middle lobe. While we can have some respiratory distress as it happens, it usually resolves fairly nicely over a day or two.
Here is some nice info on general asthma treatment to read over. I prefer starting with oral steroids and then weaning onto all inhaled meds if I can.
I hope this helps, let me know if you have any other questions.