President has left the First Lady's child (4)
I believe the answer is:
abel
'first lady's child' is the definition.
(son of Eve - the first woman)
'president has left' is the wordplay.
'president' becomes 'Abe' (Abraham (Abe) Lincoln).
'has' says to put letters next to each other.
'left' becomes 'l' (common abbreviation).
'abe'+'l'='ABEL'
'the' acts as a link.
(Other definitions for abel that I've seen before include "One man was murdered" , "He was slain by Cain" , "One of the first children" , "fellow killed by brother" , "Second child" .)