Greasy mat daughter cleared (4)
I believe the answer is:
oily
'greasy' is the definition.
(I know that greasy can be written as oily)
'mat daughter cleared' is the wordplay.
'mat' becomes 'doily' (I have seen 'Decorative mat ' mean 'doily' so perhaps 'mat' could also mean 'doily').
'daughter' becomes 'd' (genealogical abbreviation).
'cleared' is a deletion indicator.
'doily' with 'd' taken away is 'OILY'.
(Other definitions for oily that I've seen before include "Viscous" , "Smooth; like lubricant" , "Like a lot of fish" , "Covered in lubricant" , "Unpleasantly ingratiating" .)