Do you go to the EF or OF?
In the Canon (EF), before the Consecration, there are three prayers. One for the Church, one for the living and one to invoke the saint. Here are the English translations:
*Te Igitur: For the Church
Most merciful Father, we humbly pray and beseech Thee, through Jesus Christ Thy Son, Our Lord, to accept and to bless these + gifts, these + presents, these + holy unspotted Sacrifices, which we offer up to Thee, in the first place, for Thy Holy Catholic Church, that it may please Thee to grant her peace, to preserve, unite, and govern her throughout the world; as also for Thy servant N . . . our Pope, and N . . . our Bishop, and for all orthodox believers and all who profess the Catholic and Apostolic faith.
For the Living
Be mindful, O Lord, of Thy servants and handmaids N . . . and N . . . and of all here present, whose faith and devotion are known to Thee, for whom we offer, or who offer up to Thee this Sacrifice of praise for themselves and all those dear to them, for the redemption of their souls and the hope of their safety and salvation: who now pay their vows to Thee, the everlasting, living and true God.
Communicantes: To Invoke the Saints
In communion with, and honoring the memory in the first place of the glorious ever Virgin Mary Mother of our God and Lord Jesus Christ; also of blessed Joseph, her Spouse; and likewise of Thy blessed Apostles and Martyrs, Peter and Paul, Andrew, James, John, Thomas, James, Philip, Bartholomew, Matthew, Simon and Thaddeus, Linus, Cletus, Clement, Sixtus, Cornelius, Cyprian, Lawrence, Chrysogonus, John and Paul, Cosmas and Damian, and of all Thy Saints. Grant for the sake of their merits and prayers that in all things we may be guarded and helped by Thy protection. Through the same Christ our Lord. Amen.*
Just fill in the blanks for the living, the Pope and the bishop.
After the Consecration, there are a series of other prayers (still EF), asking for grace etc. And a prayer for the dead which is as follows:
Be mindful, also, O Lord, of Thy servants and handmaids N . . . and N . . . who are gone before us with the sign of faith and who sleep the sleep of peace. To these, O Lord, and to all who rest in Christ, grant, we beseech Thee, a place of refreshment, light and peace. Through the same Christ our Lord. Amen.
Again, just fill in the blanks.
Also, offering a Mass or a chaplet of the rosary etc., for a specific intention is all just about that… your intentions. You can offer a Mass simply be praying, before Mass, “O Lord, I offer this Mass to thee for [list your intentions]”.