The Holy Rosary of the Blessed Virgin is an excellent way of praying for a special intention. Neither our Faith nor its devotions are mechanicalistic. There’s not a specific way in which one would go about offering the entire Rosary or a certain mystery for special intentions. Traditionally, each mystery has been given a particular intention corresponding in some way for which one might pray. For example, the Agony in the Garden might be prayed for an increase in contrition for sins or the Scouring for purity, etc.
Remember, we pray the Rosary primarily to meditate on the life of Jesus that we may receive His graces and grow closer to Him. Secondly, we constantly, through the prayers to Mary, ask her intercession, for she is more righteous than us. “The prayer of a righteous man has great power in its effects” (James 5:16). Mary’s prayers to Christ are more efficacious than ours because she is the greatest and most-pure and holy of all God’s creatures.
We should ask Mary, our Queen, to pray for our intentions, offering them to Christ, that He might grant them to us. Her prayers have great power. When you pray the Rosary for an intention, simply tell Jesus and Mary your intention, either before praying or at the beginning of the mystery. For example, pray, “Dear Jesus, through the prayers of your holy and spotless mother, our glorious Lady and Queen, the Mother of God and Ever-Virgin, Mary, I ask the following…” or “Mary, my mother, I ask this XYZ, that you would, through your prayers, present my intentions to your Son Jesus, our True God.”
Whatever works for you do all for the greater glory of God.
In caritate Christi et Mariae,
David