She was legitimised by her father the King of Sardinia (the Duc de Saint Simon, always enraged by the elevation of bastards, was very scathing about that) and married in 1714 to her cousin, Victor Amadeus Prince de Carignan. He was a very close relative of the King and at one stage treated almost like his heir but he was so extravagant that his father-in-law became fed up with supporting him and in 1717 cut off his allowances.
The Prince then moved incognito to France (under the name of the Comte de Bosco), where he was later joined by the Princesse and where they claimed the Soisson inheritance which had been confiscated when Savoy had become an enemy of France under Louis XIV. The Prince de Carignan was given a court appointment (Intendant des Menus-Plaisirs - a sort of Master of Ceremonies) and entered into the louche life under the Regent - "the hotel of L'hôtel de Madame la Princesse de Carignan was one of the most dangerous for gambling in the capital". The Princesse eventually developed a close relationship with Cardinal Fleury ("She pretends to be devout and makes money out of the transactions of the court through the Cardinal, with whom she is on good terms"). However, she was also an intimate of the Duc de Bourbon, Louis XV's prime minister after the death of Phillipe d'Orleans. These connections were important as the Prince de Carignan proceeded to incur massive debts in France to add to those already contracted in Savoy (he eventually died a bankrupt in 1744) so it was important to keep in with the great officers of the court of France.
The Princesse de Carignan reported the intrigues of Versailles to her father at length, including the first and last attempt of Queen Marie to try to influence Louis XV politically - which involved the Duc de Bourbon's trying to oust Cardinal Fleury, a move which ended very badly for the Duc de Bourbon. However, the Princesse sided with Fleury, and when the Duc de Bourbon suggested via an intermediary that if she tried to mend the relationship between himself and the Cardinal that he would pay off her husband's huge debts in both France and Savoy and settle an income of half a million livres on her, she indignantly refused. Although the Queen sought her advice as to how to reconcile with the King, who had been very offended by Marie's attempts to interfere between himself and Fleury, the Princesse de Carignan never became very intimate with the Queen and in 1726 she and Fleury were speculating about who would replace Marie if she should die in childbed (they rather fancied Princess Charlotte of Hesse-Rheinfels, sister of the Princesse de Carignan's half-brother's wife, who in 1728 became the wife of the Duc de Bourbon as of course Marie did not die in childbirth but had 10 children and outlived the Princesse de Carignan by a couple of years).
In 1763 Leopold Mozart wrote in a letter that "today my little girl was given a small, transparent snuff-box, inlaid with gold, from the Princess Carignan, and Wolfgang a pocket writing case in silver, with silver pens with which to write his compositions; it is so small and exquisitely worked that it is impossible to describe it".
She had four or five children by the Prince de Carignan, of whom two survived: Luigi Vittorio (who married yet another Princess of Hesse-Rheinfels and became the father of the Princesse de Lamballe, amongst others) and Anna Teresa, who married as his second wife, Charles de Rohan, Prince de Soubise (she died at the age of 28 and the Prince de Soubise married yet another Princess of Hesse-Rhinefels).
And that's all I know.