What is the ruling on praying during the Adhan, and should the prayer be repeated if the Adhan for Asr is called during the Dhuhr prayer?

Delaying prayer beyond its prescribed time without a valid Shar'i excuse is a grave sin. If someone misses a prayer, or sleeps through it, or is heedless of it until its time expires, they are obligated to perform it immediately upon remembering it. This is based on the Prophet, peace and blessings be upon him, who said: "If one of you oversleeps or is heedless of a prayer, let him pray it when he remembers it, for Allah says: 'And establish prayer for My remembrance' [Taha: 14]." Therefore, your prayer is valid and you are not required to make it up.