The dua (supplication) that is recommended to be recited after the adhan (call to prayer) is known as the dua after adhan. It is a beautiful and meaningful prayer that Muslims can recite to seek blessings and ask for the intercession of the Prophet Muhammad (peace be upon him) on the Day of Judgment.
The dua after adhan is as follows:
"Allahumma Rabba hathihi dawatit tammah, was salatil qa'imah, ati Muhammadan al wasilata wal fadilah, wab'ath-hu maqamam mahmudan allathee wa'adtah, raghmatan lil 'alamin."
Translation: "O Allah, Lord of this complete call and of the prayer to be offered, grant Muhammad the privilege (of intercession) and also the eminence, and resurrect him to the praised position that You have promised, and grant him preeminence which You have promised. Send him (to be) a mercy to the worlds."
Reciting this dua after hearing the adhan is a recommended practice, as it is mentioned in the hadith (sayings and actions of the Prophet Muhammad). It is a way to acknowledge the call to prayer and to ask for the Prophet's intercession on the Day of Judgment.
Additionally, reciting the dua after adhan is a form of remembrance and gratitude towards Allah for the opportunity to engage in prayer and to seek His forgiveness. It is also a way to reaffirm our faith and belief in the finality of the Prophethood of Muhammad (peace be upon him).
It is important to note that while this specific dua is recommended, it is not obligatory. Muslims are free to offer any other supplications or make personal prayers after hearing the adhan. The dua after adhan serves as a guideline and is a manifestation of love, reverence, and respect for the Prophet Muhammad (peace be upon him).