Fag end or start of sponge bath (4)
I believe the answer is:
stub
'fag end' is the definition.
(I've seen this before)
'start of sponge bath' is the wordplay.
'start of' suggests taking the first letters.
'bath' becomes 'tub' (synonyms).
The initial letter of 'sponge' is 's'.
's'+'tub'='STUB'
'or' is the link.
(Other definitions for stub that I've seen before include "Wee bit of pencil, eg," , "Extinguish the cigarette on a counterfoil" , "Truncated remnant" , "Put out" , "Knock (e.g. toe)" .)