Assalamualaikum ustaz. When the adhan is called for Friday prayer, there are some congregants who perform the sunnah prayer. Is it permissible for a person to continue performing a sunnah prayer when the adhan is being called? Hope for an explanation.
Waalaikumussalam wrt. wbt.,
Alhamdulillah, praise and gratitude to Allah SWT for His countless blessings for us all. Praise and salutations to our beloved Prophet Muhammad PBUH, his family, companions, and all those who follow his footsteps until the Final day.
Tahiyyatul Masjid is a sunnah prayer performed by everyone who enters the mosque before he sits down. This is according to a hadith from Abu Qatadah al-Sulami RA, where Rasullullah PBUH said:
إِذَا دَخَلَ أَحَدُكُمْ الْمَسْجِدَ فَلْيَرْكَعْ رَكْعَتَيْنِ قَبْلَ أَنْ يَجْلِسَ
“When one of you enters the mosque, he should pray two rak’as before sitting down.” 
Imam al-San’ani explained this hadith saying there is a restriction against sitting when a person enters a mosque before performing a two rakaat prayer. 
Imam al-Nawawi stated that scholars have agreed that tahiyatul masjid prayer is sunnah and if there isn’t any excuse, then it is makruh to sit before performing the two rakaat prayer. 
Coming back to the above question, the related rulings should first be understood.
First, waiting until the adhan is completed for a person who just enters the mosque for Friday prayer is so that he will get the rewards of answering the adhan. Thus, it is sunnah to stay quiet and listen to the adhan as well as answer the adhan that is called out by the muezzin. 
This is based on a hadith, Rasullullah PBUH said:
إِذَا سَمِعْتُمُ النِّدَاءَ فَقُولُوا مِثْلَ ما يَقُولُ الْمُؤَذِّنُ
“When you hear the Adhan repeat what the Mu’adhdhin (the call-maker) says.” 
Second, the rulings pertaining sunnah prayer. In the context of sunnah prayer on a Friday, there are two types of prayers. First is the tahiyytaul masjid sunnah prayer and second the mutlak sunnah prayer. For those who arrive early at the mosque, it is sunnah for them to perform the tahiyyatul masjid sunnah prayer and perform as many mutlak sunnah prayers as they wished. Furthermore, the makruh ruling for sunnah prayer when the sun is situated on top of the head at noon is also abrogated due to the great advantages of Fridays.
Sunnah tahiyyatul masjid prayer is a two rakaat prayer performed by anyone who enters a mosque before he sits. This follows the hadith:
فإِذَا دَخَلَ أَحَدُكُمُ الْمَسْجِدَ فَلاَ يَجْلِسْ حَتَّى يُصَلِّىَ رَكْعَتَيْنِ
“When one of you enters a mosque, he must not sit until he prays two Rak’ah.” 
Tahiyyatul masjid is automatically considered as being performed when a person performs any obligatory or sunnah prayers. Its purpose is so that a person wouldn’t sit in a mosque before first performing any prayer to honour the mosque. 
This prayer isn’t abrogated even if a person enters the mosque and stands for a long time. This is stated in Sabil al-Muhtadin written by Syeikh Arsyad al-Banjari. He said: “Hence, its sunnah isn’t abrogated by standing even if it is for a long time.” 
The same applies to Fridays, if a person enters a mosque when the imam is already at the pulpit or delivering his sermon, it is sunnah for him to perform a brief tahiyyatul masjid prayer. Imam al-Syarwani in Hasyiyah of Tuhfah al-Muhtaj (2/455) by Imam Ibn Hajar al-Haitami said:
“And the makruh tahrim through ijma’ for a person who attends the Friday prayer starts if a performs sunnah prayer when the khatib is at the pulpit and sits there, even if he can’t hear the sermon (because he is far or others). The exception is only for tahiyyatul masjid sunnah prayer for those who has just entered the mosque when the khatib is already at the pulpit. Thus, it is sunnah for him to perform the sunnah tahiyyatul masjid prayer and it is obligatory for him to make it brief. This is if he has performed Friday qabliah sunnah prayer. If not, then it is sufficient for him to perform only two rakaat and this is enough for him to receive the sunnah tahiyyatul masjid without adding another two rakaat.”
This is also in accordance with a hadith from Jabir RA, he said Rasullullah PBUH said:
إِذَا جَاءَ أَحَدُكُمْ يَوْمَ الْجُمُعَةِ وَالإِمَامُ يَخْطُبُ فَلْيَرْكَعْ رَكْعَتَيْنِ وَلْيَتَجَوَّزْ فِيهِمَا
“When one of you comes on Friday while the imam is preaching, he should pray two rak’as and make them short.” 
If the khatib has yet to end his sermon, he should wait to perform the obligatory prayer. If a person immediately sits without first performing the two rakaat prayers, then it is abrogated and it is invalid if he performs it then. Moreover, it is obligatory for him to remain seated and listen to the sermon until it is time to perform the Friday prayer.
However, in our opinion, if a person enters a mosque when the second adhan is being called, the situation is different from when a person enters when the imam is delivering the sermon. We are inclined to state that it is prioritized for him to immediately perform the tahiyyatul masjid prayer and leave the advantage of answering the second adhan. This is to celebrate the advantages of listening to the sermon from start to finish which is prioritized over answering the adhan. This is as stated in madhhab Syafie.
 Narrated by al-Bukhari (444) and Muslim (1687)
 See Subul al-Salam, 2/62
 See al-Majmu’ Syarh al-Muhazzab, 4/52
 See Fiqh al-Manhaji, 1/117
 Narrated by al-Bukhari (586) and Muslim (383)
 Narrated by al-Bukhari (433) and Muslim (714)
 See Fiqh al-Manhaji, 1/215
 See Sabil al-Muhtadin, 2/15
 Narrated by Muslim (875)
 See Fiqh al-Manhaji, 2/210