Claimed falsely to have tucked a note inside the cover (4)
I believe the answer is:
lied
'claimed falsely to have tucked a' is the definition.
I can't judge whether this definition defines the answer.
'note inside the cover' is the wordplay.
'note' becomes 'e' (musical note).
'inside' means one lot of letters goes inside another.
'the cover' becomes 'lid' (lid is a kind of cover).
'e' placed into 'lid' is 'LIED'.
Can you help me to learn more?
(Other definitions for lied that I've seen before include "conveyed a false impression" , "Stated falsely" , "Was untruthful; song" , "Song in Berlin" , "piece of music" .)