Woman offering half-truth (4)
I believe the answer is:
vera
'woman' is the definition.
(a woman's name)
'half-truth' is the wordplay.
'half' suggests halving the letters.
'truth' becomes 'veracity' (I've seen this before).
'veracity' halved is 'VERA'.
'offering' is the link.
(Other definitions for vera that I've seen before include "__ Lynn, the forces' sweetheart" , "Girl's name (meaning faith or true)" , "Dame -- Lynn, singer" , "& 16 'The Forces' Sweetheart'" , "and 15: Singing Dame" .)