According to the excellent research of the generous John HAYSOM of Bristol (UK), at this forum, I have seen the following which should help you (see 6th generation below, with possible alternative spelling of DAMER/DAMON):
1st generation. Robert HAYSOM born circa 1646, occupation Purbeck Marbler. Robert was one of the signatories to the newly reaffirmed Articles of the Order of Purbeck Marblers in 1698. On 27 Apr 1680 he appears to have married Bridget (born 1646), the widow of John GILL. This John GILL died in 1673 but a second John GILL was also a signatory to the 1698 articles.
The Order is basically like the association of "free" (i.e. independent) miners that once flourished in the Forest of Dean, Gloucestershire. The members of the Order still meet each Shrove Tuesday in Corfe Castle to exercise their ancient rights.
Robert may have therefore entered the Order of Marblers by marrying a quarrier's widow. Bridget later died and was buried on 16 Jan 1694. Robert married secondly on 25 Apr 1695, in Swanage Parish Church to Mary KATE, who in turn died and was buried 20 Oct 1732 at Swanage. Robert died circa 1736, Swanage, being buried 6 Jan 1736?
Robert was one of two HAYSOMs (the other being Thomas - his brother?) who appear in the Swanage parish registers in the late 17th century.
Robert HAYSOM had three children by Mary KATE: i. John HAYSOM born 1696. ii. Robert HAYSOM born 1699. iii. Mary HAYSOM born 1699, Swanage
2nd generation. John HAYSOM born 1696, baptised on 31 Mar 1696, Swanage Parish Church, married 19 Jun 1720, in Swanage Parish Church, Margaret ILES
John and Margaret had issue including their eldest son: 3rd generation. Christopher HAYSOM, marbler, born 1720 (died 1779 from injuries after fall from cliff in Durlston Bay) and Katherine WARREN
Christopher and Katherine had issue including: 4th generation. William HAYSOM born 1753 and first wife Sarah TOMES (or spelt TOMBES)
William and Sarah had issue including: 5th generation. Isaac HAYSOM (born 1776, baptised 27 Dec 1776 Swanage Parish Church) married 1800 in Swanage Parish Church to Jane COOPER.
Isaac and Jane had issue including: 6th generation. Isaac HAYSOM born 1803, baptised 31 Jul 1803, Swanage Parish Church, married Elizabeth DAMER, born 1803, Swanage, died 1 Aug 1869, Swanage. Isaac died 27 Oct 1880 Swanage; Elizabeth and Isaac both buried Swanage Old Cemetery. Isaac and Elizabeth had issue: i. Charles HAYSOM born 1834; baptised 1834, Swanage Independent Chapel. ii. John HAYSOM born 1837.
7th generation. This younger son John HAYSOM born 1837, Quarryman, married Martha [unknown], born 1833, Durweston [sic - Durleston?], Dorset. John alive in 1891 at Swanage.
Children of John and Martha were: i. Elizabeth HAYSOME born 1875, Swanage, occupation [taken from 1891 census?] Dressmaker's Apprentice. ii. Annie HAYSOME born 1876, Swanage.
8th generation. John does not record if Elizabeth and Anne had children so I would be interested to know if you descend from either of these two.
Your family is distantly related to my wife's family. They, however, descend from John HAYSOM (born 1696) and Margaret ILES's *youngest* son, John, who was born in 1734; via another John Haysom born 1765; then a George HAYSOM born 1789; and in turn a Susannah Haysom born 9 Dec 1816. Susannah was my wife's 3xgreat-grandmother who married (secondly) Henry GOULDING in Winchester in 1846 (and so on …).
I had quite a bit more detail from John, in fact. You perhaps may wish to contact him directly. (His email has changed from the one shown on this forum, though.)