Should I pray Tahiyat Al-Masjid upon entering the mosque while the Mu'adhdhin is calling the Adhan, or should I follow the Adhan then pray it?

It is preferred to follow the Mu'adhdhin (caller to prayer) and then perform Tahiyyat al-Masjid (greeting of the mosque prayer), so that the reward for both acts of worship is achieved. However, on Friday, it is preferred to hasten with Tahiyyat al-Masjid in order to listen to the Khutbah (sermon), because listening to the Khutbah takes precedence over following the Mu'adhdhin.

Based on the detailed explanation of Sheikh Ibn Uthaymeen, if the Adhan (call to prayer) is for Friday prayer (the Adhan that is recited before the Khatib), then hasten to perform Tahiyyat al-Masjid because listening to the Khutbah is obligatory and responding to the Mu'adhdhin is not obligatory. However, if the Adhan is for a prayer other than Friday prayer, then it is better to respond to the Mu'adhdhin and make Du'a (supplication) after the Adhan, and then perform Tahiyyat al-Masjid.