Sailor's beginning with a bucket initially and mop (4)
I believe the answer is:
swab
'mop' is the definition.
(synonyms)
'sailor's beginning with a bucket initially' is the wordplay.
'beginning' indicates taking the first letters.
'with' becomes 'w' (abbreviation).
'initially' suggests taking the first letters.
The first letter of 'sailors' is 's'.
The initial letter of 'bucket' is 'b'.
's'+'w'+'a'+'b'='SWAB'
'and' acts as a link.
(Other definitions for swab that I've seen before include "Small piece of cotton used to cleanse a wound" , "Hospital cleaner" , "Small pad for surgical use" , "Wound-cleaner" , "Surgical cleaner" .)