Existing by begging, having lost a horse (5)
I believe the answer is:
being
'existing' is the definition.
('be' can be a synonym of 'exist')
'begging having lost a horse' is the wordplay.
'having lost' suggests deleting specific letters.
'a horse' becomes 'gg' (geegee or GG is a childish term for a horse).
'begging' with 'gg' taken away is 'BEING'.
'by' is the link.
(Other definitions for being that I've seen before include "Human, perhaps" , "Existence" , "Existing" , "substance" , "A matter of 9 in the play" .)