Is there any specific Du'a (supplication) to be said during the pause between the two parts of the Friday Khutbah (sermon)? Is there any Prophetic Hadith regarding this?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

Abu Moosa Al-Ash’ari narrated that he heard the Prophet saying about the hour on Friday when the Du'aa’ (supplication) is undoubtedly answered: “It is the period between the time the Imam sits on the Minbar (pulpit) and the time the prayer finishes.” [Muslim]

An-Nawawi believes that what is correct about the hour on Friday (when the supplication is undoubtedly answered) is what is narrated in the Hadeeth above. There are over 40 different opinions about this question. But the above opinion is the sound one. So, it is desirable for the people led in prayer to seize the opportunity to supplicate when the Imaam is sitting down between the two stages of the Khutbah. But we do not know any special Du'aa’ to that.

Nonetheless, we can ask Allah for anything in this life or the Hereafter. However, one should not supplicate as long as the Imam is delivering his sermon although he can say Ameen when the Imam supplicates.

Allah knows best.