Back in 2007 there was an exchange about Penelope West, wife of the Herbert Pelham who died in 1624:
https://groups.google.com/g/soc.genealogy.medieval/c/GHukRPSybUU/m/77IsTvVzOWEJ. I apologise if what follows partially or wholly duplicates posts in subsequent threads.
Maddison's Lincolnshire Pedigrees states that Herbert Pelham married twice, his second wife being an Anne who was buried at Boston, Lincolnshire on 20 March 1637/8; and that Herbert and Anne had four children: Anthony, baptised at Boston on 24 May 1621; Anne, buried at Boston on 10 March 1622/3; Martha, baptised at Boston on 6 March 1622/3 and buried there on 19 April 1624; and Elizabeth, baptised ast Boston on 6 July 1624 and living unmarried in 1674. See A R Madison, Lincolnshire Pedigrees, Viol. III, Harleian Society, 1904, p. 767,
https://archive.org/details/lincolnshireped00larkgoog/page/n32/mode/2up.
Douglas Richardson, in both the 2nd edition of Magna Carta Ancestry (Vol. IV, p. 327) and in Royal Ancestry (Vol. V, p. 355), states that Penelope died in about 1619. This may be on the basis of the baptism dates of Herbert Pelham's daughters by his supposed second wife.
Another researcher has been looking at relevant records of estate administrations at a Family History Centre and she and I have been having discussions. The information she has found proves that Penelope was alive when Herbert Pelham died, and hence that he did not have a second wife called Anne.
On 24 September 1624 Penelope was granted administration of the estate of "her late husband Herberte Pelham late of Boston above said deceased". The administration bond was also signed by another Herbert Pelham and by Anthonie Pelham. The signatures of all three are clearly visible on the record. Penelope was therefore very much alive then. - Inventory and Bond of Harbart Pelham senior of Boston, Wills and administrations of the Episcopal Consistory Court of Lincoln, England, 1506-1857, Administrations 1624 number 132, Familysearch reference Film 008315772 image 435. (The record is unfortunately, not viewable online but the researcher who found it has shared images with me.)
Co-signatory Herbert was probably her son, who was born in about 1600 (his 1626 marriage licence gives his age then as 26 - "Allegations for Marriage Licences issued by the Bishop of London, 1611 to 1828", Vol. II, Harleian Society, 1887, p. 177,
https://archive.org/details/publicationshar09socigoog/page/n184/mode/2up. Anthonie (d. 1659) was likely to have been Penelope's nephew, son of her sister Elizabeth who was the 2nd wife of Penelope's father-in-law.
There is an administration record for the estate of the Anne Pelham buried on 20 March 1637/8: she was a spinster and administration of her estate was granted to her brother Charles.
Joseph Lemuel Chester in an 1879 article in the New England Historical and Genealogical Register states that, according to a Roll of his ancestry by Herbert Pelham's great-grandson Samuel Bennett, Penelope and Herbert Pelham had 16 children, of whom one was Anthony, almost certainly the Anthony baptised in 1621. - "Herbert Pelham, his Ancestors and Descendants", NEHGR, Vol. XXXIII, 1879, pp. 289-290,
https://books.google.co.uk/books?vid=0L8Iquw0p__lPIgsgILkBfq&id=jtYSJeIFBFAC&pg=RA1-PA290&lpg=RA1-PA290&dq=herbert+Pelham&as_brr=1&redir_esc=y#v=onepage&q=herbert%20Pelham&f=false.