Neatly trim thick hair covering current dog (8)
I believe the answer is:
manicure
'neatly trim' is the definition.
(to manicure is to neatly trim one's nails)
'thick hair covering current dog' is the wordplay.
'thick hair' becomes 'mane'.
'covering' indicates putting letters inside.
'current' becomes 'i' (symbol for electric current).
'dog' becomes 'cur' (cur is a kind of dog).
'i'+'cur'='icur'
'mane' enclosing 'icur' is 'MANICURE'.
(Other definitions for manicure that I've seen before include "Grooming for 10, 17 and 18" , "Hands/nails treatment" , "Trim" , "Work on hand" , "Cosmetic treatment for the hands" .)