An interesting, but easy, question to answer.
There was a Matthew Trewhella married to Grace UREN in 1836 but he and his family were at Ludgvan in 1841 and the names William and Catherine did not appear among their children.
He was related to the Matthew of this query but was also about 30 years younger.
The Matthew Trewhella at Phillack (of this query) was baptised at Towednack 14th March 1779 s/o William TREWHELLA and Ann (nee COCK) and was the brother of my 4xgt-grandfather.
Matthew married Martha DANIELL at Towednack 2nd August 1801.
He was buried at Phillack 26th March 1859 (age 80, of Riviere - a farm in Phillack Parish).
Martha was buried at Phillack 15th November 1853, age 72, fof Riviere.
Martha was baptised at St Ives 3rd October 1781 (IGI) d/o John and Margaret (nee STEVENS) DANIEL.
In 1841 there were three children of Matthew and Martha living at home.
These were, as mentioned in the query, William, John and Catherine.
There were seven further children of Matthew and Martha of whom two Eliza (1809-1809) and Martin (1818-1819) were deceased prior to the 1841 Census.
Of the remainder:-
Matthew (1802-1849) was married and living elsewhere
Martha (1804-
?) was married to Samuel Michell of Towednack
Christopher (1813-1875) never married but was 'of Riviere Farm' when he died and left his Will.
James (1815-1884) married his cousin Ann TREWHELLA in 1840 and resided at St Erth.
Margaret (1823-
?) married John HARVEY at Phillack in 1855.
Of the three children 'at home' in 1841:-
William (1807-1877) married Eliza Ninnes SAUNDRY at Madron in 1843 and resided at Trenerth Farm in Gwinear where he died.
John (1811-1883) married Jane CHINN in 1850 and resided at Treglissian (Treglistian?) Farm in Phillack where he died.
Catherine (1820-1897) married Samuel OULD at Phillack in 1848.
To my knowledge, so far, the BAWDEN people living nearby had nothing to do with this family and, having a quick look at the entries in the 1841 Census, I can see no tangible connection.
This does not mean, however, that it was not the case.
Another 'cousin' of Matthew Trewhella (without racking the brain too much - I think was a first cousin) was Martin TREWHEELA.
(Please ignore the spelling of the surname in some of these entries as I have produced them as they appeared in the Parish Registers).
This Martin TREWHEELA marred Margaret THOMAS at Towednack in 1783.
Their daughter Margaret married Edward BAWDEN at Ludgvan 15th May 1813 and had, to my knowledge, seven children.
Edward BAWDEN was, I understand, baptised at Phillack 8th September 1792 s/o Edward BAWDEN and Elizabeth (nee COCK).
Edward Bawden was, at the time of his marriage, 'of Gwinear'.
Hope this helps.
Look forward to further input and conversation on this family and hope I may be able to help further.