Untruth provided by female willingly (4)

I believe the answer is:

lief

'willingly' is the definition.
(I've seen this before)

'untruth provided by female' is the wordplay.
'untruth' becomes 'lie' (lie is a kind of untruth).
'provided by' says to put letters next to each other.
'female' becomes 'f' (abbreviation).
'lie'+'f'='LIEF'

(Another definition for lief that I've seen is " In old usage, gladly or willingly - file away".)

I've seen this clue in The Telegraph.

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