What is the ruling on someone who prayed Fajr believing the time had begun, then it became clear to him during the prayer that the call to prayer (adhan) was being made? Does this necessitate expiation or repentance?

The fundamental principle for determining the beginning of prayer times is the sharia-compliant signs. Many contemporary scholars have issued fatwas permitting the use of reliable prayer timetables. Therefore, if you rely on a trustworthy timetable, your prayer is valid, and hearing the adhan is not a prerequisite. The delay in the adhan you heard might be due to the muezzin. It is legislated to interrupt one's prayer upon hearing the adhan from the mosque to attain the virtue of congregational prayer.