It is quite common for babies to be born with one (or both) tear ducts not yet fully open. Tears cannot flow normally and accumulate and become thick and "sticky".
The problem (whether you put antibiotics or not), may recur several times until the canal opens (of its own accord, often before 6 months but sometimes 1 year or even more).
It is therefore a temporary problem, without danger, but disturbing
In the meantime, you just have to wash your eyes well and gently massage the corner of your eye (on the side of your nose, after washing your hands, to help the canal to unblock and the bag to drain), at least 3 times a day, for example with each drink and to be patient...
Conjunctivitis?
Although it is less likely, it can also be conjunctivitis. Conjunctivitis can occur at any age, even in a newborn. This is why newborns are often given a small antibiotic ointment as a preventive measure because their eyes may have come into contact with bacteria in the vagina during birth. The eye (the white part) will then be more red and possibly swollen. You should apply warm water compresses 4 times a day and eventually consult a doctor.