I engaged in foreplay with my wife before the adhaan of Fajr, without having intercourse, but maniy (semen) came out of me approximately three or four minutes after the adhaan; is my fast valid or do I have to make up that day?
The fast is valid. The fact that semen came out after the adhaan of Fajr does not affect the fast, because it is maniy that resulted from permissible intimacy (at a permissible time).
The scholars (may Allah have mercy on them) mentioned a similar matter, which has to do with one who has intercourse at night, then maniy comes out of him by day. They said: his fast is valid.
An-Nawawi (may Allah have mercy on him) said: If he has intercourse before Fajr, then withdraws when dawn breaks or immediately after dawn breaks, and ejaculates, it does not invalidate his fast, because it resulted from permissible intimacy, so nothing is required of him (by way of making up the fast).
End quote from al-Majmoo‘ (6/350)
Muhammad ad-Dasooqi (may Allah have mercy on him) said: If he has intercourse at night, and emits semen after dawn, what appears to be the case is that nothing is required of him, like one who applies kohl at night then the kohl reaches his throat during the day.
End quote from Haashiyat al-Dasooqi (1/524).
And Allah knows best.