Not sure if my advise is a help to you.
My boi also have phlegm and intend to see doctor but my mother in law asked her friend a nurse, she says if go see doctor(at polyclinic) they will not provide any medicine one because he's still very young and such phlegm maybe it's due to when he's just born the hospital didn't extract all the phlegm 100% clean that's why it's still there.
Try to feed her with "chin feng san" so that she can pass out those phlegm with her shit (the nurse teach us one)
Actually i've tried and it seems to work on my son.
Give it a try?
Chin feng san is also another thing that can take every night on your finger and rub insider her mouth to let her sleep better.(old ppl say)