Adhan Du'a - دعاء الأذان

Du’a between adhan and iqamah
With regard to the time between the adhan and iqamah, du’a is encouraged at this time and it is mustahabb (preferred).
Anas (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “Du’a is not rejected between the adhan and iqamah, so engage in du’a (supplication).” (Narrated by al-Tirmidhi, 212; Abu Dawud, 437; Ahmad, 12174 - this version narrated by him. Classed as sahih by al-Albani in Sahih Abi Dawud, 489).
There is a specific du’a to be said immediately after the adhan:
Jabir ibn ‘Abd-Allah (may Allah be pleased with him) reported that the Messenger of Allah (peace and blessings of Allah be upon him) said: “Whoever says when he hears the call to prayer : ‘Allahumma Rabba hadhihi’l-da’wat il-tammah wa’l-salat il-qaimah, ati Muhammadan il wasilata wa’l-fadilah, wab’athhu maqaman mahmudan illadhi wa’adtah (O Allah, Lord of this perfect call and 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),’ my intercession for him will be permitted on the Day of Resurrection.” (Narrated by al-Bukhari, 589)