What colour is the gunk? If clear/white/yellowish it won't be an infection. If its green/brown it could be, but still may not. If you've not got other symptoms/temperature, infection is unlikely. It may be a reaction to the medication, worth a discussion with the respiratory nurse IMO.
I'd book an appointment with the nurse/doc. They may give you a pot to cough & spit in for a lab test. If it's been 'literally' weeks should get it checked.
You need to continue with your meds including rescue pack, but also contact your GP and tell him you would like to have your sputum tested. This will indicate which exact antibiotic you need, get it and take it. This is the normal path to recovery.
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