Holy father saddled little horse without tail falling out (7)
I believe the answer is:
pontiff
'holy father' is the definition.
(I know that holy father can be written as pontiff)
'little horse without tail falling out' is the wordplay.
'little horse' becomes 'pony' (a pony is a little horse).
'without tail' means to remove the last letter.
'falling out' becomes 'tiff' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'pony' with its last letter taken off is 'pon'.
'pon'+'tiff'='PONTIFF'
'saddled' is the link.
This may not be correct. It may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for pontiff that I've seen before include "Title for a pope" , "High-priest" , "Religious leader" , "Head of the Roman Catholic church" , "Bishop" .)