Thanks for the query to H.C.M. Forum.
There are two separate disorders.
Burping relates to stomach ( gastric area disorders).
Phlegm from lungs.
Since you haven't mentioned about smoking history.
Cough is there accompanied with expectoration ( phlegm) , reason is allergy with super added infection ( may be smokers cough, bronchitis
, lung abscess
Almost one year passed after endoscopy
for once again gastroscopy will help you.
Consult physician and get in complete examination as X ray of chest ( phlegm) and gastroscopy .
After this deal accordingly.