Snoop around back of slate quarry (4)
I believe the answer is:
prey
'quarry' is the definition.
(I know that quarry can be written as prey)
'snoop around back of slate' is the wordplay.
'snoop' becomes 'pry' (I've seen this in another clue).
'around' indicates putting letters inside.
'back of' says to take the final letters.
The final letter of 'slate' is 'e'.
'pry' placed around 'e' is 'PREY'.
(Other definitions for prey that I've seen before include "Hunted creature" , "Animal's quarry" , "Animal hunted by another for food" , "Quarry, target" , "Animal hunted or caught for food" .)