Rugs put round hall originally for this subject (5)
I believe the answer is:
maths
'this subject' is the definition.
(maths is a kind of subject)
'rugs put round hall originally' is the wordplay.
'rugs' becomes 'mats' (I've seen this before).
'put round' means one lot of letters goes inside another.
'originally' indicates taking the first letters.
The initial letter of 'hall' is 'h'.
'mats' placed around 'h' is 'MATHS'.
'for' is the link.
(Other definitions for maths that I've seen before include "Science of magnitude and number" , "Study of numbers and sums" , "Subject involving figures (abbr)" , "Algebra, geometry, calculus etc (abbr)" , "The science of number, quantity and space" .)