Did we err in praying Fajr after the Adhan on the mobile phone by approximately 10 minutes, with the possibility that the Adhan was before the actual time of the prayer, and is re-performing the prayer obligatory upon us?

Prayer is not permissible except with certainty or strong presumption of the entry of the time. Whoever prays while doubting, their prayer is invalid and must be repeated, even if it later becomes clear that it coincided with the correct time. However, whoever prays based on a strong presumption of the entry of the time, their prayer is valid unless an error becomes apparent to them.