How authentic is the Hadith: "The Messenger of Allah, peace and blessings be upon him, would seek forgiveness for the believing men and believing women, and the Muslim men and Muslim women, every Friday"?

This hadith is weak due to the weakness of its chain of narration, which includes unknown narrators. However, the principle of supplication during the khutbah (sermon) is generally permissible, not with the specification of a particular wording or a fixed time. Rather, the Khatib (preacher) should supplicate with what is appropriate for the condition of the people and the country, while avoiding fixed formulas that are not mentioned in the Sunnah. This is because if the hadith is not authentic, then the supplication should not be taken as a regular Sunnah to be constantly observed.