Going over border, bad time to make journey (10)
I believe the answer is:
'journey' is the definition.
(I know that pilgrimage is a type of journey)
'going over border bad time' is the wordplay.
'going over' is a reversal indicator.
'border' becomes 'lip' (I've seen this in another clue).
'bad time' becomes 'grimage' (I can't explain this - if you can you should give a lot more credence to this answer).
'lip' in reverse letter order is 'pil'.
'to make' is the link.
Can you help me to learn more?
(Other definitions for pilgrimage that I've seen before include "Religious traveller's journey" , "special journey?" , "What Chaucer,s travellers were engaged upon" , "Holy journey" , "Religiously-inspired journey" .)