Medicine’s short — not a good time for hajj (10)
I believe the answer is:
pilgrimage
'for hajj' is the definition.
(Hajj is the Muslim pilgrimage to Mecca)
'medicine's short not a' is the wordplay.
'medicine' becomes 'pill' (pill is a kind of medicine).
'short' means to remove the last letter.
'not a' becomes 'grimage' (I can't explain this - if you can you should give a lot more credence to this answer).
'pill' with its last letter taken off is 'pil'.
'pil'+'grimage'='PILGRIMAGE'
'good time' is the link.
This may not be correct. Some or all of it may belong to another bit of the clue.
(Other definitions for pilgrimage that I've seen before include "Religious traveller's journey" , "What Chaucer,s travellers were engaged upon" , "journey to shrine" , "Journey to a sacred place" , "special journey?" .)