See also my 'Alternative History of Britain: The Hundred Years War' (Tim Venning, pub. Pen and Sword, 2013) page 109 and ibid, 'The Wars of the Roses' (Pen and Sword 2013) p. 9, where I briefly ,mention the possibilities arising from either Duke John of Bedford (d 1435) or Duke Humphrey of Gloucester (d 1447) having legitimate sons. As in this article, I also infer that Humphrey's heirs would have been in a weaker position re: the English succession, as his first marriage, to Jacqueline of Holland, took place in dubious circumstances affected by her disputedly legal divorce from her first husband - a close relative and ally of England's ally Duke Philip of Burgundy - which was later invalidated by the Pope under Burgundian pressure. This would have bastardised any son of Humphrey's by her, and his second marriage , to Eleanor Cobham, was disliked by many nobles and was open to attack politically by his enemies as she had been a low-born lady in waiting and so was not of the usual 'acceptable' elite class for a great noble/prince's wife; it was thus easier for his enemies to get this marriage invalidated too.
So if Humphrey had been insisting in the 1440s that after himself his son , by either woman, was his heir to all his lands (which in OTL went to the impoverished and financially needy King and the allegedly extravagant Queen as he had no legal heir) and thus to the Crown too, this would be open to attack whether or not he was arrested and ruined and/or died in 1447. The Beauforts as well as York's faction could have targeted Humphrey's heir. A son of Bedford by his OTl wife Anne of Burgundy would presumably have been born in the mid-late 1420s and not been adult enough to succeed Bedford in his role as regent of/ commander in France in 1435, meaning that that role went to Humphrey and then to York as in OTL, and even in the early 1440s a young and untried son of Bedford's would have been at a disadvantage even compared to the more experienced John and Edmund Beaufort. But possibly the King and Suffolk's faction could have bought him off from joining their critics after York was recalled and sidelined from power and Humphrey was destroyed,in 1447, by giving him a command in Normandy - though by that date the lack of men or money would have meant that he could not do much. Had he put up a more respectably aggressive defence there in 1449-50 than Edmund Beaufort did in real life, this could give him support after the end of English Normandy in 1450 and make him a potential rival to York - if he was so inclined, eg by bitterness at the Beauforts or their ally Suffolk not giving him enough men and money to hold out longer. Or anger at them being too keen to hand over Maine to France on easy terms but keep this shabby deal secret from the English public in 1445-8?
Bedford married Anne in June 1423; a son of his by his second wife Jacquetta of St Pol (by her next marriage the mother of Edward IV's Queen, Elizabeth Woodville, and about a dozen other children), married April 1433, would have been in an even poorer position until c. 1450 as too young to take part in war or politics. After 1453, when Henry VI had his first outbreak of a catatonic stupor, they would have been in a good position to challenge York as the reversionary heir, or once his son Edward was born later in 1453 as regent; and as Henry's death would not mean York becoming king, this might have eased the attitude of fear and hatred to York by the Queen and her (Beaufort-led) faction and/or stopped anti-Beaufort nobles all gravitating to York's side. Also, if the crises had developed as in OTL and led to York seizing power as 'regent' in 1455 (as older and more experienced than Bedford's son?) and then being attacked and forced to flee for his life to Ireland by the Queen in 1459, York would not have been able to claim the throne on his return in 1460. A different dynamic to the so-called Wars of the Roses and a three-way fight between the Queen's party, 'the second duke of Bedford', and York?
Bedford's son would have been much stronger in political terms, or been a senior military commander in France if competent enough, had Bedford (born 1389 and so a late 34 at marriage) married earlier - eg if his father Henry IV had married him off in his international schemes of the late 1400s, and/or his intervention in France to back up the 'Armagnac' faction against the Burgundians in 1411-12. At this point, with the future Henry V the heir but at odds with his father and favouring Burgundy instead, Bedford was however not the next heir, to whom the King would turn if at odds with his eldest son - this was Henry IV's second son Thomas, Duke of Clarence (b 1388, k 1421). The King could still have arranged a good, probably French marriage to an heiress to lure Bedford to support him. Clarence, killed by the French and Scots in battle at Bauge in March 1421 in OTL, left no children by his 1411 marriage to the elder John Beaufort (the King's illegitimate half-brother)'s widow Margaret Holland; but as said, a son of this marriage (born c. 1412-16?) would have been in a good position to be a senior commander in France after 1435 and , as adult, would also be the next heir to Henry VI ahead of Humphrey so this might make Humphrey less of a threat to the King and reduce the chances of his arrest in 1447. Humphrey was however naturally truculent and quarrelsome, as seen by his long feud with Bishop Henry Beaufort (the next brother of the elder John Beaufort), and he even undermined his brother Bedford in the mid-1420s by marrying Jacqueline of Holland and so annoying Bedford's vital ally Philip of Burgundy.
The possibility also arises of a son of Bedford (if born before c. 1418) or of Clarence succeeding as full viceroy / governor of Normandy and the rest of 'English France' (with Paris to 1436) after Bedford died in 1435, though only as regent - ie with sovereign powers - until Henry VI's majority in 1437. Then if they were not given adequate troops or money to fund them , the English position would collapse in the mid-later 1440s as in OTL, once Charles VII had got a proper army organised and funded and had put down his feuding nobles in the major 1440 revolt; or if 'Bedford II' was less aggressive and more cautious the abortive peace-conference of 1439 at Gravelines could have established a truce that kept the status quo for a few more years but was later broken by a reinvigorated France. (As the truce proposals included France insisting that any full peace would require Henry VI to abdicate as king of France and to be a French vassal duke of Normandy and Aquitaine, and H VI always refused this, then no long term peace was viable.) So could 'Bedford II' be evicted from France by or soon after 1450, be furious at the lack of support sent by Henry VI - mainly due to incompetent planning and systemic over-spending plus a corrupt court - and turn on his cousin the King in a different sort of civil war to OTL? And end up as 'King Henry VII' or 'King John II' with the - unreliable - backing of York, provided that he had no son to succeed him and he gave this heirship to York?