Is expiation obligatory for someone who swore by Allah that their son would not do something, then excepted by saying: "Only for today, and this is if Allah wills," after a slight pause?

The exception in an oath is only valid if it is directly connected to the oath, unless the separation is due to an irresistible excuse (such as a cough or a sneeze). Consequently, the intention to make an exception after the oath has concluded and silence has fallen upon it is not valid, and expiation becomes obligatory if the sworn-upon act occurs.