His mother left his father soon after Henry's birth, accusing him of infidelity and residing in a convent. They reconciled two years later, but Maria Klementyna died in 1735, ten years before her sons made the great attempt to retake the throne in 1745.
When that attempt failed, Henry returned to Rome and began his ecclesiastical career during the reign of Pope Benedict XIV, progressing through the minor orders while a Cardinal-Deacon in 1747 and the sub-deaconate in 1748 and then being ordained deacon and priest in 1748. He became the Cardinal-Bishop of Frascati, a diocese near Rome in 1761. He was quite wealthy with many properties and benefices, but he lost it all during the French Revolution and in support of Pope Pius VI, who was imprisoned by Napoleon and died in August of 1799. The College of Cardinals, with Cardinal Henry Stuart met in exile in Venice to elect Pope Pius VII in the contentious papal election of 1800.
In the meantime, when his brother Bonnie Prince Charlie died in 1788, the Cardinal had become the de facto Pretender, but he did not seek the throne and was not recognized by the Papacy, being called the Cardinal Duke of York. His priesthood and cardinalate really made it impossible for him to seriously be considered a candidate, anyway. While his brother had been roaming throughout Europe, even visiting England and renouncing his Catholicism (briefly) for any advantages in his attempt to regain the throne, Henry Cardinal York's path probably disappointed the Young Pretender, especially as he had no legitimate heirs.
After Napoleon's fall, the Cardinal Duke of York returned to Frascati and became Dean of the College of Cardinals in 1803. He died on July 13, 1807 when he was 82 years old. The line of the Pretenders was taken up by Charles Emmanuel IV of Sardinia who was the grandson of Charles II's favorite sister, Henrietta Anne, wife of Louis XIV's brother Philippe of France, Monsieur. Charles Emmanuel IV never actively sought the throne of England.
Some notes about minor orders: The minor orders, suppressed by Pope Paul VI in 1972 after the Second Vatican Council, were Acolyte, Exorcist, Lector and Ostiarus/Porter. They were called minor orders because they were not part of the Sacrament of Ordination and the seminarians who filled their ceremonial roles were not required to be celibate, yet. The Porter was the lowest office; the Acolyte the highest. Note that the Exorcist assisted only with the Rite of Exorcism that is part of the Sacrament of Baptism. Two minor orders survived the suppression as ministries, open to the laity: acolyte and reader, although I have never seen those ministries practiced as outlined in Pope Paul's decree. Note also that in the orders now dedicated to the celebration of the usus antiquior (for example the Priestly Fraternity of St. Peter and the Institute of Christ the King Sovereign Priest) these minor orders and the sub-diaconate are permitted for their seminarians.