I found one on FreeBMD for Guisborough district but GRO Index had it as Lofthouse, which is now known as Loftus, a town in the Guisborough Registration District. As far as I know, Lofthouse/Loftus has never been a Registration District.
With use of Union etc it appears that the new index may be utilising exactly what was written at the top of the page at the time the Book was filled in so if 'wrong' the error may well have been made 100-150 years ago (assuming it is not a modern transcription entry from illegible handwriting on the page).
Lofthouse was a Sub-District within Guisborough, though generally we only come across these when we see the paper or pdf certificate.
The official paper version would say
Registration District Guisborough / Birth in the Sub-District of Lofthouse.The old page would look something like the attached from a pdf last week. In this case the
Superintendent's Registrar's District Warwick should be typed into the computer, not the District
Radford.
Registration County : Yorkshire North Riding.
Created : 1.7.1837 (originally "Guisbrough").
Abolished : 1.10.1936 (to become part of Cleveland and Whitby registration districts).
Sub-districts : Danby, Guisborough (Guisbrough), Kirk Leatham,
Lofthouse (Loftus), Marske, Skelton.
https://www.ukbmd.org.uk/genuki/reg/districts/guisborough.htmlSometimes the Superintendents District name was identical to the Sub District name eg Births in the Sub-District of Southam in the Superintents District of Southam. Only way to tell for sure if the error was made then or now is to order the pdf but I'd probably put my money on the the computer entry. Can see me doing that after had spent an entire day inputting.