What is the ruling on the muezzin’s saying, after the Adhan: “May Allah’s salat and peace be upon our master Muhammad and upon his family and all his companions,” and is it considered an innovation (bid‘ah)? And what is the permissible formula for sending blessings upon the Prophet after the Adhan?

If the muezzin says "The Messenger of Allah has spoken the truth" loudly, immediately following the adhan (definition), this is an innovation (bid‘ah), because the adhan concludes with "There is no god but Allah." However, sending blessings upon the Prophet (peace and blessings be upon him) quietly after the adhan has concluded is legislated for both the muezzin and the listener, due to the Prophet's (peace and blessings be upon him) saying: "When you hear the muezzin, say as he says, then send blessings upon me."