Is expiation obligatory for someone who swore to abandon masturbation for a month, then broke their oath after 22 days, with doubt as to whether they said "in sha' Allah" after the oath with a time interval?

An exception in an oath is only valid under certain conditions: including the exception being directly connected to the oath, and the intention of making the exception existing before the completion of the oath. Doubt about the existence of an exception does not remove the ruling of expiation, unless it is the person's habit to make exceptions.