Answered by Ustadh Tabraze Azam
Question: [1 ] Should the adhan be made at the START of the time when the five prayers becomes permissible, or can it be delayed in order to better fit the timings of the congregational prayers (e.g. making adhan late for Fajr, as its congregational prayer is usually delayed in the Hanafi madhab)?  Also, the dua which we say after adhan (.. اللَّهُمَّ رَبَّ هَذِهِ الدَّعْوَةِ التَّامَّةِ), should we only say it after really having heard the adhan, or can it also be said when the adhan “should” have been made according to the time, in an area were adhan can’t be heard.
Answer: Assalamu alaikum wa rahmatullahi wa barakatuh,
I pray that you are in the best of health and faith, insha’Allah.
 There would no harm in delaying the call to prayer (adhan) until the recommended times for prayer.
 The supplication is normally said after the call to prayer (adhan). If the call to prayer (adhan) is usually given in one’s neighbourhood and one cannot hear it, it would be recommended to give the call to prayer (adhan) oneself — though, it need not be long nor loud. Thereafter, one can say the supplication.
[Haskafi, al-Durr al-Mukhtar; Shurunbulali, Maraqi al-Falah/Tahtawi, Hashiyat al-Tahtawi]
And Allah alone gives success.
Checked & Approved by Faraz Rabbani