Colouring that’s unknown, established for use in pudding (8)
I believe the answer is:
dyestuff
'colouring' is the definition.
(a dyestuff changes something's colour)
'unknown established for use in pudding' is the wordplay.
'unknown' becomes 'y' (algebra).
'established' becomes 'est' (abbreviated used in shop signs).
'for use in' is an insertion indicator.
'pudding' becomes 'duff' (duff is a kind of pudding).
'y'+'est'='yest'
'yest' placed into 'duff' is 'DYESTUFF'.
'that's' is the link.
(Other definitions for dyestuff that I've seen before include "Substance used in staining" , "Colouring substance" , "Anil perhaps" , "Colourant" , "Colouring agent" .)