There is no true proof that Edward was the son of Kenelm but it appears to be so.
Winslow Families of Worcestershire, 1400-1700 by Brandon Fradd pub. Newbury Street 2009 - Fact and Fiction
Fiction 2: - KENELM WINSLOW owned Newport's Place and Clerkenleap. Again, the chart in the Register in 1867 makes this statement. A much earlier publication by Thomas Habington in his history of Worcestershire states that he had "heard" that KENELM WINSLOW bought and sold Newport's Place, but he was not at all sure that it was true. 
Fact: - We see from the court rolls of Kempsey that KENELM WINSLOW was the tenant farmer, not the owner, of Newport's Place. Dr. Treadway Nash, whose family bought Clerkenleap, states in his history of Worcestershire that "the Winslows" owned Clerkenleap and sold it around 1650 to John Nash. It might be expected that a historian whose family owned the property would know something more about its origins, but Dr. Nash was not more specific. We note below that JOHN WINSLOW of Kempsey paid taxes on "Bramallo" in 1571 and that Broomhall was a messuage in Clerkenleap. This JOHN WINSLOW was of Draycott.
The chart in the Register further states that KENELM WINSLOW's grandson, RICHARD, sold Clerkenleap in 1650. So far we have not encountered any evidence for such a RICHARD WINSLOW. All the grandchildren we believe to have been descended from KENELM WINSLOW - namely, Gov. EDWARD and his brothers and sisters - had left Worcestershire long before 1650. However, there was definitely a RICHARD WINSLOW of Draycott who died in 1659. The value of his father's inventory in 1615 was more than £110, whereas KENELM WINSLOW's was about £70. The difference could have been easily explained by a large messuage at Clerkenleap owned by the Winslows of Draycott. Until further information is found, we believe that it was the Winslows of Draycott who owned Clerkenleap and sold it to the Nash family.
Fiction 5: - The Winslows of Kempsey descend from the Winslows of Earls Croome.
In an article in the Register in 1967, John Hunt pointed out that in the three and one-half centuries since the first two Winslows came to America in the Mayflower, "not a few conflicting statements concerning their origin and status, including several wrong pedigrees, have gotten into print" He then proceeded to provide yet another wrong pedigree, namely: GALFRID WINSLOW of Earls Croome, County of Worcester, 1425; RICHARD of the same, 1430; WILLIAM of the same, 1471; RICHARD of the same, will dated 1546; THOMAS of Kempsey; KENELM of Kempsey and Worcester. Hunt cited a manuscript in the possession of D. KENELM WINSLOW as basis for these claims. (151)
Fact: - We have shown that the Winslows of Kempsey can be traced in Kempsey to 1432 and that the court rolls of Earls Croome do not exist prior to 1500, even though there is a document indicating the existence of GEOFFREY [GALFRID] WINSLOW of Earls Croome in 1425. It is possible that D. KENELM WINSLOW had access to court rolls for Earls Croome now missing, but if so, he did not publish their details in his book, Mayflower Heritage. (152)
Furthermore, as the record shows, there were three RICHARD WINSLOWS before 1546: two in Kempsey (who were probably closely related), and one in Earls Croome. The RICHARD WINSLOW of Earls Croome (from which all the pedigrees derive the later Winslows of Droitwich) must be discarded as a possible fit for the Richard posited by Hunt.
Fiction 6: - The Winslows of Kempsey descend from THOMAS WINSLOW who married Eleanor Peacock in 1539. Once again, this was a statement by D. KENELM WINSLOW.
Fact: The court rolls of Kempsey do suggest a THOMAS WINSLOW as the ancestor of the Winslows of Kerswell, but this is a suggestion at best, not a direct statement. There is no THOMAS WINSLOW in the ancestry of the Winslows of Draycott. The probable ancestor of KENELM WINSLOW of Kerswell was RICHARD WINSLOW, who was already of age in the 1540s and so could not be the product of a marriage registered in Severn Stoke in 1539.
 Survey of Worcestershire, 2:148. Speaking of Kerswell, he says "the heyres of John Clopton heald in the same manor and thease 3 last I thincke weare severall tenauntes of the same landes, the inheritance as I have heard of Sir Richard Newport, and sold by him to Kenelm Winslowe and by Winsloe to Sir John Bucke; and which in the white book of the bishopricke is as I guesse called Newport's place. Nowe for Newport's place I am certain; at the rest, I rove." As shown below, Thomas Newport held the manor of "Careswell" in Worcestershire and granted it to his son Richard in the marriage contract with Richard's wife, but that Kenelm Winslow was the tenant farmer, not owner. It was sold not to Buck, but to Humphrey Baker before 1622.
The Five Winslow Brothers of Early Massachusetts
PROBABLE DESCENT OF THE WINSLOW BROTHERS FROM HENRY WINSLOW OF KEMPSEY
The male-line ancestry of the five Winslow brothers appears to be as follows, based on the records presented in this book:
Henry Winslow, of Kempsey (by 1411-1451). John Winslow, of Kempsey (by 1436-after 1480).
Thomas Winslow, of Kerswell in Kempsey (by 1469-aft. 1536). He appears to have been the father or grandfather of:
Richard Winslow the Younger, of Kerswell (aft. 1510-fl. 1544-56). Kenelm Winslow, of Kerswell and Clifton (ca. 1550-1607).
Edward Winslow, b. ca. 1570, m. 1594 Magdalen Oliver; they were the parents of the five Winslow brothers of Massachusetts.
Will of Kenelm Winslow:
"In the name & feare of God Amen-the xiiith daye of Aprill in the year of our Lord 1607, I Kenelme Wynslowe of the cittye of Worcester, Yeom¹, being of verye pfect memorye although sicke in bodye doe make & declare my last Will & Testamt in maner & forme following viz! First I comend my Soule to the Eternall God & my bodye to the Earth to be buryed in comelye sort of burial after my decease.
Item. I devise and appoint vs in money to the pore of the prshe of St. Andrew wherein I dwell to be distributed by my wife or by her appointmt. And as touching my goods and chattels I will & appoint the custodye thereof (my funerall's discharged & my debts paide) to Katherine my very loving Wife whom I ordaine constitute & appoint to be my Sole Executrix of this my prnt Will & appointing & wishing her nott to alter the pptie thereof (things overworne excepted) wthout the consent of my eldest Sonne whom I require to be a guide & comforter to her. And such of my household stuffe as she shall thinke well of I license her to dispose of to such of my children & grandchildren as shall best please her and the same nott to be delivered until after her decease and then the same to be delivered to them as the guift & legasie as well of me to them as of my saide wife. These being Witness prnte att the publishing hereof by me-Kenelme Wynslo-John Evayns-Edward Tovy+-Richard Calwall his marke "H."
Proved at Worcester on the 9th day of November 1607 by Katherine (the Relict of the deceased the sole Executrix." from W.S. Appleton's papers.
¹A common man; the first or most respected class; a freeholder; a man freeborn; a yeoman in England is considered as next in order to the Gentry. Gentry - in Great Britain are classes of people between the nobility and the common people.