White's beginning to get grey in laundry (4)
I believe the answer is:
wash
'laundry' is the definition.
('wash' can be a synonym of 'laundry')
'white's beginning to get grey' is the wordplay.
'beginning to' says to take the initial letters.
'get' is a charade indicator (letters next to each other).
'grey' becomes 'ash' (ash grey is a kind of grey**).
The initial letter of 'whites' is 'w'.
'w'+'ash'='WASH'
'in' acts as a link.
(Other definitions for wash that I've seen before include "Of animals, cleanse by licking" , "Stand up" , "Clean; thin coat of paint" , "Clean with water" , "bath" .)