I know I am coming into this very late!
There was a marriage on 04 Jun 1807 at St. Nicholas, Whitehaven, of William SPENCE, weaver & Abigail BURNS, Widow. Witnesses: George BLENKINSOP, John BRIGGS.
This is from a transcription of the parish registers by Cumbria FHS.
I could not find the burial of a William BURNS at St. Nicholas between 1804 and 1807. Nor indeed was there a burial of a William BURNS in Nev Ramsden's Copeland Index (an index of all the parish registers of the Copeland area of Cumberland up to 1837) between 1804 and 1807. So perhaps he died at sea?
But there is the burial of an Abigail SPENCE, age 34, on 17 Feb 1813 at St. Nicholas, Whitehaven. No further details given in the parish register.
This would suggest a birth for Abigail of around 1778/79.
And also, the Copeland Index has the marriage on 17 May 1802 at Cleator of William BURNS & Abigail NEWTON.
Also from the Copeland Index: Baptism on 19 Feb 1780 at St. Bees: Abigail NEWTON, daughter of James & Betty of Harras, husbandman.
That is the only Abigail NEWTON baptism in the Copeland Index around the right time, with a very wide margin.
As regards baptisms for a William BURNS, I found the following possible ones in the Copeland Index:
Bap. 07 Jun 1779 at Harrington: William BURNS, son of Thomas
Bap. 19 Dec 1784 at Harrington: William BURNS, son of Thomas & Rachel (although this is a little late for a marriage in 1802)
The last baptism of a William BURNS before 1779 was in 1760, so rather unlikely.
There was also the baptism of a William BURN, son of George of Workington, on 07 Jan 1774 at St. Michael, Workington - but he did not have an S at the end of the surname, and Workington does not fit very well either.
It is however possible that William was baptised in the Presbyterian Chapel, or that indeed he was born in Scotland.
Good hunting!
Petra