If it's really not going away, ask for some anti-biotic drops. DS had a sticky eye for weeks, and everyone kept saying it was a blocked duct and to clean with boiled water and breastmilk, which we did with no result.
Eventually, the GP presribed drops and it cleared up in a couple of days.
my ds had this for 6 mths it was blocked tear duct just wiped with warm water on cotton wool
it is normal for blocked duct to take months to clear as it says if not cleared after so much time then duct will need to be cleared opened what ever it was now was 4 yrs ago but after 6 mths roughly it cleared
My ds2 is 6wks old and has this. Some days it completely goes then comes back. He had his 6wk check yesterday and the doctor took a swab to check if he needs antibiotics. He just said to clean it with cooled boiled water and cotton wool. Will get swab results tomorrow.