For newborns (or kids below 2), it's best to just see a paediatrician because there's very little (if at all) medication that they can take and it can be dangerous at times - especially when their breathing is blocked by phlegm or mucus. I don't really have any remedies for you, but seeing a PD might be your best bet :)
How old is your baby? Mine also have a lot phlegm and I tried many methods till my gal friend recommended me to see TCM...
Kris Poh