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' indicates named letters should be taken away.
'a horse' becomes 'gg'.
'begging' with 'gg' removed 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" .)