Lied about taking the old cover (6)
I believe the answer is:
eyelid
'cover' is the definition.
(I have seen 'Protective cover' mean 'eyelid' so perhaps 'cover' could also mean 'eyelid')
'lied about taking the old' is the wordplay.
'about' indicates an anagram.
'taking' indicates putting letters inside.
'the old' becomes 'ye' (resembles a historical spelling of 'the').
'lied' with letters rearranged gives 'elid'.
'elid' going around 'ye' is 'EYELID'.
(Other definitions for eyelid that I've seen before include "Ocular shutter" , "It's used in blinking" , "Covering for organ of sight" , "Where stye may occur" , "Skin for blinking" .)