I haven't identified this yet. Presumably it is the name of a house, named after Thornhaugh in Cambridgeshire. It is mentioned in several family history documents (below) as "the Aunts' home". This probably means the spinster sisters Elizabeth Barton (b.1832) and Emily Barton (1842-1927), who presumably moved there with their guardian 'Aunt Rick', who died in 1892 (her funeral was at Shirley just east of Croydon). Whoever lived there, they were evidently there as early as October 1890 and as late as Christmas 1902.

Mentions

It appears as a note by Lucy Barton (1808-1903) on a letter from Maria Done to John Barton:
"Letters from my Grandmother to her husband - my Grandfather whose portrait they have at Thornhaugh L.F.G." [LFG stands for Lucy's married name 'Lucy FitzGerald']

*Memoirs of John Barton:
"Of my grandfather, son of the inventor of the spinning jenny, we know but very little, but that little shows him to have been a man both of cultivated tastes and also of a very philanthropic spirit. His portrait was painted in the year 1779 and is still to be seen in the library at Thornhaugh, having come into my Uncle Bernard’s, (the poet) possession by a cousin’s chain of circumstances, portrays him as surrounded by his books, a volume of Locke’s Essays in his hand, a flute lying on the table at his side. (Note by 'DMB': a photograph of this painting is on our attic staircase)" [I do not know who DMB is, nor when this note was added.]

*Memoirs of Emily Elliott:
"It was the beginning of this year [1877] that Pryke came to us, as cook. She was then a young widow if 27 with one delicate boy. As you all know she has been in the family almost ever since; partly with us, and partly at Thornhaugh - and is (as I write, 1903) now living with Will at Rawdon, as Housekeeper." p82

"We left Cambridge the middle of December [1893]; and spent Christmas at Croydon – some at Thornhaugh, others at Powyscout." p104

"We had spent Christmas 1902 at Thornhaugh, Croydon, the dear Aunts' home" p127

Emily's memoirs also mention:
"On October 8 or 9 [1890] we set our faces Homewards - and spent the 12th (which was the 27th Anniversary of our Wedding Day) at Coblentz on the Rhine. At Dusseldorf we left Jessie for eight months with the Gericke family to learn German; which proved to be both a happy and successful venture. We reached the dear Aunts' home at Croydon with Ethel on the 15th; and were welcomed in Cambridge again on the 17th having been absent a year and a day." p99