I looked in the newspapers for references to Long Houghton between 1830 and 1839.
The Sheffield Independent Newspaper mentions Long Houghton a few times in relation to fox hunts, for example on 15th October 1831 it says
'Lord Fitzwiliam's fox hounds will meet on. . . . Saturday the 22nd at Long Houghton.'
There is also mention of a marriage in the Saturday 21st August 1830 edition. It says
"A few days ago, at Darfield Church, Mr John Sellers, farmer of Long Houghton, near Barnsley, to Mary, the third daughter of Mr Waterton of Houghton Hall."
There are references to other Long Houghtons in the 1830s in the newspapers, including one near Alnwick, which I believe is near Berwick on Tweed, so likely a different one.
There is also reference to an estate belonging to the Duke of Northumberland, but that one is near Howick in Northumberland, so I assume a totally different one.
The only mentions which fit the Long Houghton you were talking about were in relation to fox hunts and that wedding above - which makes me believe it must have been a fairly rural place which was perhaps falling out of use as a place name.
I couldn't find a single reference to a place called 'Barnsley Cliff Bridge'.
However there were a few references to a place called Cliff Bridge near Barnsley.
Perhaps the most useful are these two.
Firstly from the Sheffield Independent on Sat 8 July, 1848
"Woman Drowned - On Sunday morning, about one o clock, a women of the name of Watts was drowned in Barnsley canal, at Cliff Bridge, when in a state of intoxication. It appears the deceased and her husband had been drinking together at Barnsley on Saturday night, and has set of at midnight to go home to Littleworth, but the husband falling in the road, the deceased was walking by herself on the canal side when she accidentally fell in. Although assistance was procured in a short time, yet the poor woman was found to be dead when she got out of the water. She was nearly 60 years of age."
So that would suggest to me that Cliff Bridge was somewhere between Barnsley and Littleworth.
There is also an advert from the same newspaper on Saturday 20th April 1839. It reads "Oaks, Hoyle Mill and Cliff Bridge Quarries near Barnsley." It gives some details of the sort of blocks you can buy from there and then reads "The two former quarries (presumably meaning Oaks and Hoyle Mill) being situate on the banks of the Deane and Dove canal and the later (presumably meaning Cliff Bridge) adjoining the Barnsley Canal afford every facility of shipping to any part of the Kingdom."
The advert also asks for quarrymen to apply for jobs, which is maybe why your relatives were born there?
There is one marriage listed in the same newspaper, on 8 July 1848, "At Rotherham, on Monday last, Mr Wm Walker, builder of Cliffbridge, near Barnsley, to Miss E. Siddall, Fitzwilliam Street, Sheffield."
There were lots of other references to a cliff bridge, but that is in Scarborough.
Hope that is helpful.
