As I pointed out recently, Alice de Lacy, countess of Lincoln, and her husband Thomas of Lancaster, earl of Lancaster and Leicester, were expecting a child in 1307 or 1308, but ultimately had no surviving children. Thomas did, however, have at least two illegitimate sons, John and Thomas of Lancaster.
John of Lancaster was said in 1349, rather fascinatingly, to be "the son of a married man and a spinster related in the third degree of kindred." There is no doubt about his identity: he was addressed as "John de Lancastria, son of the late Thomas, earl of Lancaster, scholar of theology."  I've never managed to figure out who his mother, Earl Thomas's second cousin or second cousin once removed, might have been. John gained an M.A. in theology, and by 1355 had joined the household of his cousin, Thomas's nephew Henry of Grosmont (d. 1361), first duke of Lancaster, earl of Leicester, Lincoln and Derby, father-in-law of John of Gaunt from 1359, and the maternal grandfather of King Henry IV. Earl Thomas's brother Henry, earl of Lancaster and Leicester (d. 1345), Duke Henry's father, asked the pope for a dispensation for his kinsman John of Lancaster in 1343 on account of his illegitimacy, and Edward III also acknowledged "John de Lancastria, son of Thomas, earl of Lancaster" as his kinsman in 1350. John, a canon of Uttoxeter, Lincoln and Salisbury, died in or before 1361. 
Earl Thomas's other known illegitimate son was Thomas, who originally was a knight and later became a friar, and in 1354 was called "Thomas de Lancastria, knight, son of Thomas, earl of Lancaster". He was knighted by Edward III during the king's French campaign of 1346.  It was said in 1354 that Thomas of Lancaster "passed his youth at a university and other places, and afterwards in a war", and took part in an attack on the French town of Sens. Because he had killed and wounded men there, he "wishes to change his life" and to join the Franciscan order. Thomas was "illegitimate, being the son of a married man and a mother of whom it is doubted whether at the time he was begotten she was married or a spinster."  I don't know what happened to him after 1354, or whether he had the same mother as his brother John of Lancaster, or when they were born, or which of them was the elder.
As well as Earl Thomas of Lancaster's illegitimate sons, his nephew and nieces acknowledged a family called the Walkingtons as their relatives. Henry of Grosmont made his will in March 1361, and appointed ten executors.  One was Blanche, Lady Wake (d. 1380), the eldest of Duke Henry's six sisters, and eight were men, including the bishop of Lincoln and the abbot of Leicester. The other was nostre tres chiere cosyne de Walkynton, "our dearest cousin of Walkington". Several academic historians who should really know better have identified this person as the long-term Lancastrian retainer Sir William Walkington, but there are two big problems with this. 1) William died before 7 February 1357 , and 2) it only requires a fairly basic knowledge of French to spot that chiere cosyne is the female form and cannot refer to a man. William Walkington's wife was named Eleanor, and she presumably was the person who was Henry's executor. An indenture of 1361 between Duke Henry's son-in-law John of Gaunt and four of the late duke's ten executors talks of the 'lady de Walkington'.  Eleanor Walkington held the Wiltshire manor of West Grimstead in 1339 and 1361 as the widow of John de Grymstede, and she and William were said in 1350 to be both "of the diocese of Lincoln" and "of the diocese of Salisbury". They married without royal licence before 2 July 1338. 
In petitions to the pope in 1343 and 1344, two of Duke Henry's sisters, Blanche, Lady Wake, and Maud, dowager countess of Ulster (d. 1377), acknowledged the clerk Master Robert Walkington M.A., a canon of Lincoln, York and Uttoxeter, as their kinsman. Robert had a sister named Agnes, also acknowledged as her relative by Lady Wake, who married Sir John Mauduyt of the diocese of Salisbury. In the early 1330s, Henry, Blanche and Maud's sister Isabella of Lancaster, nun and later prioress of Amesbury (d. 1348/9), sent gifts of a girdle and a silken purse to Sir William Walkington, and Robert Walkington and Agnes Mauduyt were presumably his siblings. Agnes married Sir John Mauduyt before 7 May 1328, probably around 23 February 1328, and her brother Robert Walkington was the feoffee when John granted an Oxfordshire manor to himself and Agnes jointly.  Robert Walkington was named as a clerk of Henry, earl of Lancaster, father of Duke Henry, Blanche, Isabella and Maud, in 1325 and 1342, and another likely relative was John Walkington, granted forty marks of rent annually in Staffordshire and a manor in Wiltshire by the younger Henry of Lancaster (d. 1361) in 1349. The elder Henry of Lancaster (d. 1345) gave the Gloucestershire manor of Minsterworth to Sir William Walkington for life in March 1328, and the following year William was one of the knights given a safe-conduct to accompany Henry overseas. In 1332, Henry gave Walkington another three manors in Derbyshire. 
The Lancasters held the earldom of Lincoln after Thomas of Lancaster's widow Alice de Lacy died in 1348, and they held the town of Uttoxeter in Staffordshire, as part of the territories that had once belonged to the Ferrers family, earls of Derby, and were given to Edmund of Lancaster in 1269. Earl Thomas of Lancaster's illegitimate son John of Lancaster M.A. (d. c. 1361) was a canon of Lincoln and Uttoxeter, and so was Robert Walkington M.A. Sir Thomas Mauduit (b. October 1287) was executed as an adherent of Thomas of Lancaster in Pontefract in March 1322, and Thomas's son and heir John proved his age in May 1332: he was born in Warminster ('Weremynstre') Wiltshire on 2 February 1310 ("the feast of the Purification, 3 Edward II").  This would seem to be the man who married Agnes Walkington (d. 1369), as Agnes's husband was said to be of the diocese of Salisbury and Warminster is certainly in that diocese. Walkington, incidentally, is a village in Yorkshire, near Beverley.
There are, therefore, numerous points of connection between the Walkington family and the Lancaster family, and several of Earl Henry of Lancaster's children acknowledged the Walkingtons as their kinsfolk. The big question is, though, how were the Walkingtons, people of comparatively humble birth and rank, cousins of the partly royal, hugely wealthy, well-connected and influential Lancasters? One possibility is that the Walkingtons were related to Earl Henry's children via Henry's wife, Maud Chaworth (1282-1322). Maud was the daughter and heir of Patrick Chaworth (d. 1283), himself the younger brother and heir of Payn Chaworth (d. 1279), and the Chaworth brothers were the sons of Hawise of London, who died in or before September 1274.  The genealogy of the Chaworths/Londons is rather obscure, and it is certainly possible that Maud Chaworth's children were related to the Walkingtons on her side of the family, even if the precise connection is doomed to remain elusive. Although Maud was an heiress and brought Henry of Lancaster the Welsh lordships of Kidwelly and Carmarthen and lands in five English counties, when their marriage was arranged in 1291 Henry was only his father Edmund's second son, and his older brother Thomas, Edmund's heir, was married to a much greater heiress from a much more prestigious family, Alice de Lacy. Alice's family tree is much better known than Maud's.
Another possibility is perhaps that the Walkingtons were somehow descended from the Lancasters illegitimately. As, however, I've never seen evidence that Earl Thomas or his brother Earl Henry called the Walkingtons their kinsfolk, though Earl Henry certainly showed the family great favour over the years, it seems more likely that they were related to Earl Henry's children on their mother Maud Chaworth's side. According to the family tree in one article , Payn and Patrick Chaworth had two sisters: Eva, who married Sir Robert Tybetot, and Anne, about whom no further information seems to be available, and various genealogy websites give them another sister, Emma Chaworth. Or perhaps the Walkingtons were descended from a sibling of Hawise of London.
1) Calendar of Papal Letters 1342-62, pp. 346, 357.
2) Petitions to the Pope 1342-1419, pp. 65, 193, 271, 346, 383; CPL 1342-62, pp. 346, 545, 547.
3) Petitions to the Pope, p. 262; William Arthur Shaw, The Knights of England: A Complete Record from the Earliest Time, vol. 1 (1906), p. 6 of the section called 'Knights Bachelors'; George Wrottesley, Crécy and Calais, from the Original Records in the Public Record Office, pp. 35, 209, 249, 259.
4) Petitions to the Pope, p. 262. Thomas is also mentioned in Calendar of Close Rolls 1346-9, p. 545, and Calendar of Patent Rolls 1345-8, p. 408, 487.
5) Testamenta Vetusta, vol. 1, pp. 64-6; A Collection of All the Wills Now Known to be Extant of the Kings and Queens of England, Princes and Princesses of Wales, and Every Branch of the Blood Royal, pp. 83-7.
6) CPR 1354-58, p. 506.
7) The National Archives DL 27/242.
8) CPR 1338-40, pp. 102, 356; CPR 1361-4, p. 84; Feet of Fines CP 25/1/288/46, no. 592; TNA C 143/248/13; CPL 1342-62, pp. 382, 406; CCR 1377-81, pp. 454-5.
9) Petitions to the Pope, pp. 29, 69, 74, 271; CPL 1342-62, pp. 99, 105, 145, 218, 406, 573; R.B. Pugh, 'Fragment of an Account of Isabel of Lancaster, Nun of Amesbury, 1333-4', in Leo Santifaller, ed., Festschrift zur Feier des zweihundertjährigen Bestandes des Haus-, Hof- und Staatsarchivs, Band 1 (1949), p. 492; CPR 1327-30, p. 263; CCR 1327-30, p. 365.
10) TNA DL 25/334/279; CPR 1324-7, p. 167; CPR 1327-30, pp. 258, 442; CPR 1330-34, pp. 321, 367; CPR 1348-50, pp. 282, 366, 469.
11) Calendar of Inquisitions Post Mortem 1307-17, no. 159; CIPM 1327-36, no. 479; CIPM 1365-69, no. 395; CIPM 1377-84, no. 1018.
12) CIPM 1272-91, nos. 51, 310, 477.
13) M.T.W. Payne and J.E. Payne, 'The Wall Inscriptions of Gloucester Cathedral House and the de Chaworths of Kempsford', Transactions of the Bristol and Gloucestershire Archaeological Society, 112 (1994), pp. 87-104.