Bishop put on garment over part of nun's habit (4)
I believe the answer is:
barb
'part of nun's habit' is the definition.
(barb can mean an item of headwear worn by nuns)
'bishop put on garment over' is the wordplay.
'bishop' becomes 'B' (chess abbreviation).
'put on' says to put letters next to each other.
'garment' becomes 'rab' (I can't justify this - if you can you should believe this answer much more).
'over' says the letters should be written in reverse.
'rab' in reverse letter order is 'bar'.
'b' after 'bar' is 'BARB'.
(Other definitions for barb that I've seen before include "Cutting remark" , "Sharp spike pointing backward" , "Stage whispers" , "A thorn or a gibe" , "Point of fish-hook - gibe" .)