Going through the extra small: they’re in short supply (6)
I believe the answer is:
'in short supply' is the definition.
(sparse can mean meagre or minimal)
'going through the extra small' is the wordplay.
'going through' means one lot of letters goes inside another.
'the extra' becomes 'spare' (both can mean left over).
'small' becomes 's' (abbreviation - e.g. clothes size).
'spare' placed around 's' is 'SPARSE'.
'they're' acts as a link.
(Other definitions for sparse that I've seen before include "Spares, being thin on the ground" , "Thinly dispersed" , "rarely occurring" , "Scarce like thinning hair" , "Meagre, scanty" .)