Detective left picture taken after Greek expedition (10)
I believe the answer is:
pilgrimage
'expedition' is the definition.
(thesaurus)
'detective left picture taken after greek' is the wordplay.
I cannot quite see how this works, but
'detective' could be 'pi' (private eye) and 'pi' is found in the answer.
'left' could be 'l' (common abbreviation) and 'l' is present in the answer.
'picture' could be 'image' (synonyms) and 'image' is located in the answer.
'greek' could be 'gr' (abbreviation for Greek) and 'gr' is present in the answer.
This accounts for all the letters.
This may be the basis of the clue (or it may be nonsense).
Can you help me to learn more?
(Other definitions for pilgrimage that I've seen before include "Religiously-inspired journey" , "Journey to a sacred place" , "Holy journey" , "Religiously inspired journey" , "journey to shrine" .)