بسم الله الرحمن الرحيم
It is better to respond to the muadhin and then supplicate after that, with what has been reported1, then he enters into (performing) tahiyyatul masjid (2 rakah before sitting down), however some of the scholars have made an exception when entering the mosque on the Friday and the muadhin is calling the second adhaan, (if that occurs) he (first) prays tahiyyatul masjid , so that he is able to listen to the khutbah. The reason for that is because listening to the khutbah is an obligation and responding to the muadhin is not an obligation. And safeguarding the performance of an obligation takes precedence over safeguarding (the performance of) that which isn’t an obligation.
[Source: Shaykh Al-Uthaymeens Majmoo’ al-fataawa war rasaa’il]
اللَّهُمَّ رَبَّ هَذِهِ الدَّعْوَةِ التَّامَّةِ، وَالصَّلَاةِ الْقَائِمَةِ، آتِ مُحَمَّداً الْوَسِيلَةَ وَالْفَضِيلَةَ، وَابْعَثْهُ مَقَاماً مَحْمُوداً الَّذِي وَعَدْتَهُ، [إَنَّكَ لَا تُخْلِفُ الْمِيعَادَ
Allaahumma Rabba haathihid-da ‘watit-taammati wassalaatil-qaa’imati, ‘aati Muhammadanil-waseelata walfadheelata, wab ‘ath-hu maqaamam-mahmoodanil-lathee wa’adtahu, [‘innaka laa tukhliful-mee’aad]
O Allah , Lord of this perfect call and established prayer. Grant Muhammad the intercession and favor, and raise him to the honored station You have promised him, [verily You do not neglect promises].
Reference: Al-Bukhari 1/152, and the addition between brackets is from Al-Bayhaqi 1/410 with a good (Hasan) chain of narration. See ‘Abdul-Azlz bin Baz’s Tuhfatul-‘Akhyar, pg. 38.