What is the ruling on the Friday preacher supplicating in the middle of the sermon, and the people saying "Amen" to his supplication, and his saying, "Supplicate to God while you are certain of a response," and the worshippers raising their hands in supplication?

It is recommended for the preacher to supplicate in his sermon, and for the attendees to say "Ameen" to his supplication, but without raising hands for either the preacher or the attendees, except in the prayer for rain (Salat al-Istisqa'), in which it is sunnah to raise hands, following the Prophet, peace and blessings be upon him. The preacher can supplicate in the middle or at the end of the sermon, as it is among the times of acceptance during the hour of Friday when a Muslim servant does not ask Allah for good except that He grants it to him. This hour is either from the Imam's sitting until the end of the prayer, or after Asr prayer, which is the more probable opinion.