Some priests in the Western Catholic Church will do the extra prayers after the final blessing rather than during the announcement time. I have also seen priests do extra prayers before the Mass begins or during the homily time.
One of the issues in the Western Catholic Church is that we used to have certain prayers called the “Leonine Prayers” instituted by Pope Leo that were said at the end of Mass, for certain intentions involved wiht the protection of the Church. Around the time of Vatican II, the Vatican suppressed these prayers without giving a reason. It was speculated that either the original intention for the prayers was no longer necessary, or that people might be scared by the words of one of the prayers which mentioned Satan, but no definitive answer was ever given as to why these prayers were suppressed. In recent years, a lot of priests and bishops concerned about the state of the Church have returned to saying at least one of the Leonine Prayers at the end of Mass, either at announcement time or after the final blessing.
I’ve also seen a number of priests do particular novenas, either the Miraculous Medal novena or one associated with the patron saint of the parish or sometimes St. Joseph or for a particular intention (St. Junipero Serra being invoked for vocations, for example) at Masses for a set period of time. Often the prayer is meant to help the parish or the Church in general. Often they do them after the final blessing but I’ve also seen them done during announcement time or even during homily time.