This family doesn't seem to be indexed under any of the spellings GAWTHORP, GAWTHORPE, GOLDTHORP, GOLDTHORPE on Ancestry for 1861, but I found some of them on Findmypast, where I am only able to view the transcript, not the original image.
Living in Cater Lane, Heckmondiwke:
Jonathan GOLDTHORPE, Head, Mar, 26, Farmer's labourer, b Yorks, Sandal.
Easter [sic] do, Wife, 30, Yks, Honley.
Susan do, Son [sic,] 9, Yorks, Dewsbury.
Michael do, Brother, 17, Factory Boy, do, do.
Thomas do, Brother, 15, Farmers Labourer, do, do.
Alfred do, Brother, 12, Scholar, do, do.
Mary Ann do, Sister, 8, - , do, do
There is a birth registration of Mary Ann GAWTHORP [sic] in Dewsbury RD, 1/1853. It would be interesting to know what it states regarding her parent(s).
I also found some of them in 1871, this time on Ancestry:
Heckmondwike Road, Dewsbury:
Michael GOLDTHORP, Head, Mar, 27. Wool Cleaner, Yorks, Dewsbury.
Hannah do, Wife, 32, - , Yorks, West Ardsley.
children aged 5, 3, 2, 4 months.
Alfred do, Brother, 23, Woollier, Yorks, Dewsbury.
Living next door:
Josiah GAWTHORPE, Head, 21, Railway Platelayer, Yorks, Dewsbury.
Isabella do, Wife, 20, - , Yorks, Heckmondwike.
Mary Ann do, Daur, 4 months, Yorks, Dewsbury.
Thomas not found.
Marriages in Dewsbury RD:
Josiah GAWTHORP and Isabella TOWNEND, 4/1869.
Michael GOLDTHORPE & Hannah VERITY, 4/1863.
Sam