Wa `alaykum as-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Brother, we really appreciate forwarding this question to us, and we commend your keenness on getting yourself well-acquainted with the teachings of Islam. May Allah help us all keep firm on the Right Path, ameen.
Friday is the most blessed day and Almighty Allah has bestowed many virtues on that day more than any other, so du`aa' (supplication) is something recommended on this day and especially during the khutbah (sermon) and Prayer. This is something unanimously agreed-upon by Muslim scholars and it is not allowed for a preacher to abandon it in his speech except by mistake or forgetfulness.
It was narrated that the Prophet (peace and blessings be upon him) used to pray for forgiveness for the believers every Friday during the khutbah. (Al Tabarani)
As for the virtue of du`aa' on Friday, Abu Hurairah said, “Abul-Qasim (peace and blessings be upon him) said, “On Friday there is an hour when, if a Muslim happens to supplicate at that time and ask Allah for something good, He will give it to him.” (Al-Bukhari and Muslim)
In his response to your question, Sheikh Ahmad Kutty, senior lecturer and Islamic scholar at the Islamic Institute of Toronto,Canada , stated,
Jumu`ah is the weekly assembly of Muslims; it is intended to foster a sense of community, renew their faith, and inspire them to come closer to Almighty Allah and realize the Islamic ideals in their lives; individually and collectively. The Friday khutbah is precisely intended to achieve this. It is therefore understood that the Friday khutbah should have a focused message and should follow a requisite format and content.
All Muslims agree that a proper khutbah ideally consists of praising and glorification of Allah, sending blessings on the Prophet (peace and blessings be upon him) and advising people to remain conscious of Almighty Allah through admonition and reminders based on the Qur'an and the Sunnah, and this should be followed by du`aa' for all Muslims.
Some scholars consider all of the above-mentioned as essential aspects of a Friday khutbah, while others consider some of them as simply recommended; so even though a person may be excused for having left out du`aa’ if he did so by mistake, he should not do so deliberately.
Most of the scholars belonging to both the Shafi`i and Hanbali schools consider du`aa’ as an integral aspect of Friday khutbah. Imam An-Nawawi considers this to be the authentic view of the Shafi`i school.
It is worth remembering that as mentioned in authentic traditions, Friday is the most blessed day of the week, and therefore it is doubly auspicious for dhikr and supplications. Furthermore, the Prophet (peace and blessings be upon him) said that there is a special hour on Friday when supplications are more likely to be answered by Almighty Allah; hence, it is no wonder to see that the Islamic Ummah throughout the centuries has been following the above format, and thus, they have always concluded the khutbah with du`aa’.
Based on these, it is, therefore, inadvisable for anyone to depart from this well-established practice of the Muslims. As Imam Ibn Taymiyah and others have pointed out, a scholar must never allow himself to depart from practices that have been preferred by the majority of people based on his own personal preferences, and thus, creating doubts and disquiet in their minds, especially in matters of worship.
In conclusion, it is important that preachers make use of the opportunity of the Friday khutbah to offer supplications for Muslims and to never stop doing so by following isolated opinions or practices.
Allah Almighty knows best.