NOTE: there's a much longer essay on Piers Gaveston here, and a much shorter one here.
Edward's great favourite Piers Gaveston fathered a daughter, Joan, by his wife Margaret de Clare (Edward II's niece). She was born in York in the second week of January 1312, after Piers had returned illegally from his third exile, presumably anxious about his wife and keen to see his child. Edward spent forty pounds (in modern money, hundreds of thousands of pounds) on the week-long celebration of Joan's birth in February, after Margaret's churching. Joan was only 5 months old when her father was killed.
(By the way, if Queen Isabella's first pregnancy was full-term, the future Edward III was conceived at the time of this celebration, which is one of my Favourite Facts Ever.)
The nature of Piers' and Margaret's relationship can only be a matter of speculation. They married on 1 November 1307, in the presence of the king, a few months after Piers had been created earl of Cornwall. Margaret's date of birth is unfortunately unknown, but her elder sister Eleanor was born in October 1292 and her younger sister Elizabeth in September 1295, placing Margaret's birth in late 1293 or sometime in 1294. Therefore, she was either just 14 or (more likely) 13 when she married Piers. His birthdate is also unknown, but is presumed to be about 1283, so he was about 24 at the time of his wedding.
Many people assume that Margaret 'must' have hated Piers because of his relationship with her uncle, but on the other hand, she might equally have adored him, whatever she thought was happening between her husband and uncle. As a granddaughter and niece of kings, she would have grown up in the knowledge that her marriage would be used to make or cement a political alliance, and wouldn't be a love match. Piers was fairly young, and by all accounts handsome, witty, charming and athletic (he was a star jouster), so she might easily have been dazzled by him, and at least by marrying him, she could stay in England.
She might have been angry that she was being disparaged by marrying a humble Gascon knight, but then again he was earl of Cornwall, and Edward made him one of the richest men in England. It's notable that she accompanied him to Ireland during his exile, 1308-1309, although she didn't have to - nobody would have dreamed of exiling her, the sister of the earl of Gloucester.
They only had one child that we know of, but that doesn't prove anything about how often they did or didn't sleep together. As Margaret was so young when they married, Piers may have waited a year or two to consummate the marriage. Maybe she had miscarriages. It's also important to note that Margaret only had one child by her second husband Hugh Audley too, so probably she was sub-fertile.
Joan grew up, according to her late father's wishes, at the priory of Amesbury in Wiltshire. This is often portrayed these days as shunting the daughter of a dead embarrassment into an obscure convent, out of sight out of mind, in a Victorian kind of way. Nothing could be further from the truth - Amesbury was extremely fashionable at this time, after Eleanor of Provence (widow of Henry III and Edward II's grandmother) took the veil there in 1284. Edward's sister Mary (born 1279) was a nun there, as was his niece Joan de Monthermer, and the future prioress was Isabella of Lancaster, niece of the earl of Lancaster and Hugh Despenser the Younger. Other royal women were educated there, including Eleanor de Bohun, another niece of Edward II who later married the earl of Ormonde, and possibly others too. It was a privilege to grow up at Amesbury, not a punishment or an embarrassment.
Joan died in January 1325, around the time of her 13th birthday, still unmarried, although Edward II had arranged a very prestigious marriage for her with John de Multon (born 1308), son of Lord Egremont and eldest grandson of the Earl of Ulster. Thomas's aunt Elizabeth was queen of Scotland and another aunt, Matilda, was the widow of the Earl of Gloucester (Margaret de Clare's brother). Thomas was therefore first cousin to the future Earl of Ulster William de Burgh, and also first cousin to the future Earls of Louth, Kildare, and Desmond, as well as first cousin to the children of Robert, King of Scotland, by his second wife.
Edward II paid Joan an extremely generous allowance of a hundred marks or 66 pounds a year, at a time when 40 pounds was the minimum qualification for knighthood. Even if he didn't have time to see her very often, he didn't forget about his favourite's child. (Edward was very generous after Piers' death, giving Margaret a huge allowance and taking her into his own household, as he also did many of Piers' servants.)
Piers also fathered an illegitimate daughter called Amie, mentioned in a document of the 1330s when she married John de Driby. She is described as the daughter of Piers Gaveston, and was apparently a damsel in the household of Queen Philippa, wife of Edward III. As with Edward II's illegitimate son, the mother of Amie is unknown, as is Amie's (even approximate) birth date. Amie has descendants alive today.
There was a huge amount of debate on Amie online, several years ago. Anyone interested in Piers' daughter should try googling 'Amie Gaveston' or 'Amie de Gaveston'' (with Amy and Gavaston as occasional alternative spellings). Bizarrely, it was postulated that Amie was the illegitimate daughter of Margaret de Clare, not Piers! There was also a theory that the 'Piers Gaveston' mentioned was not THE Gaveston, but another one. It's possible, but as there's no evidence at all of another Piers Gaveston, it seems a bit of a strange argument to me. Another very silly theory was that Joan was not Piers' daughter either, but fathered by Some Other Man (a similar argument to the 'paternity of Edward III' nonsense). Edward's expensive celebration of Joan's birth was explained away as Edward's wish to protect Piers from the public humiliation of being cuckolded. Honestly, I don't know where people get these ideas from.
There seems to be no reason at all why Piers couldn't have fathered children, so searching for ways in which Amie and Joan might not have been his children strikes me as silly and pointless.
The most recent biography (2003) of Edward II, by Roy Martin Haines, states that Piers Gaveston had a sister called Amie, who was with him during the siege of Scarborough in 1312. Given medieval naming methods, this makes it 99% certain that Amie was indeed Piers' daughter. Likewise, if we could find a woman connected to Edward II with a father or brother called Adam, we'd almost certainly discover the identity of Adam's mother.