Losing heart, one's ridden the tube (4)
I believe the answer is:
hose
'the tube' is the definition.
(hose is a kind of tube)
'losing heart one's ridden' is the wordplay.
'losing heart' means to remove the middle letters.
'one's ridden' becomes 'horse' (animal which may be ridden).
'horse' with its centre taken out is 'HOSE'.
(Other definitions for hose that I've seen before include "Stockings, etc." , "plumbing connection" , "One with doublet; a tube" , "Stockings; drench" , "footwear" .)