Father is on the endless trail (4)
I believe the answer is:
path
'trail' is the definition.
(I've seen this in another clue)
'father is on the endless' is the wordplay.
'father' becomes 'pa' (pa can mean dad or father).
'is on' means one lot of letters go next to another (in a down clue, letters appear on others).
'endless' means to remove the last letter.
'the' with its final letter removed is 'th'.
'pa'+'th'='PATH'
(Other definitions for path that I've seen before include "Footway or track" , "Track made by walking" , "Skin on top of head" , "Route, course" , "Pedestrian way" .)