He engaged in foreplay with his wife during the night in Ramadan, then some maniy came out of him after the adhaan of Fajr

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.