In a sense, the Church does indeed recognize a "reconciliation of desire"—perfect contrition. The Catechism of the Catholic Church defines contrition as "sorrow of the soul and detestation for the sin committed, together with the resolution not to sin again" (CCC 1451).
For contrition to forgive mortal sins it must arise out of our love of God. "When it arises from a love by which God is loved above all else, contrition is called ‘perfect’ (contrition of charity). Such contrition remits venial sins; it also obtains forgiveness of mortal sins if it includes the firm resolution to have recourse to sacramental confession as soon as possible" (CCC 1452).