What is the ruling on the muezzin saying aloud after the Adhan: "O Allah, send prayers upon Muhammad and upon the family of Muhammad. O Allah, Lord of this perfect call and the prayer to be offered..."?

Praying upon the Prophet, peace and blessings be upon him, after the Athan is a recommended Sunnah, for it has been narrated from the Prophet, peace and blessings be upon him: "When you hear the Mu'athin, say as he says, then send blessings upon me, for whoever sends blessings upon me once, Allah will send blessings upon him tenfold. Then ask Allah to grant me Al-Wasilah, for it is a station in Paradise that is not befitting except for one of Allah's servants, and I hope that I am he. And whoever asks Allah to grant me Al-Wasilah, my intercession will be due to him."

As for raising the voice with it, some scholars disliked it, while others considered it permissible as long as there is no exaggeration. Ibn Taymiyyah held that the Sunnah in supplication and sending blessings upon the Prophet, peace and blessings be upon him, is to do so in a low voice, unless there is a reason that permits raising the voice. Ibn Hajar Al-Haytami mentioned that raising the voice with it is permissible without dislike if there is no exaggeration; rather, it is a Sunnah. Based on this, exaggeration in raising the voice with blessings upon the Prophet, peace and blessings be upon him, is not legislated, and it is preferable to send blessings silently.