Two entries with the name Florence.
Florence became a Knight of the Round Table. With his brother Lovel, he joined tha party that helped Mordred and Agravaine try to trap Lancelot with the Queen. All but Mordred were killed by the escaping Lancelot.
Malory also mentions a Sir Florence in Arthur’s Continental campaign against the Emperor of Rome.
Then ... the king called Sir Florence, a knight, and said to him they lacked victual. And not far from hence be great forests and wood, wherein be many of mine enemies with much bestial: I will that thou ... go thither in foraying, and take with thee Sir Gawaine my nephew [and others].
The foragers ran afoul of a force of Spanish knights, defeat them, and win plenty of rich spoils.
Although Malory’s internal chronology is confusing, he puts the war against Emperor Lucius much too early in the work to make me feel comfortable identifying this Florence with Gawaine’s second son; morever, it seems unlikely that Arthur would put Gawaine’s son, rather than Gawaine himself, to charge of an expidition which included Gawaine.
Spain | The Legend of King Arthur
Le Morte Darthur | Sir Thomas Malory, 1469-1470