Nuisance that is taking detectives to find killer (9)
I believe the answer is:
pesticide
'killer' is the definition.
(thesaurus)
'nuisance that is taking detectives' is the wordplay.
'nuisance' becomes 'pest' (I've seen this before).
'that is' becomes 'ie' (short for Latin 'id est', 'that is').
'taking' means one lot of letters goes inside another.
'detectives' becomes 'CID' (abbreviation for Criminal Investigation Department).
'ie' enclosing 'cid' is 'icide'.
'pest'+'icide'='PESTICIDE'
'to find' acts as a link.
(Other definitions for pesticide that I've seen before include "Insect killer" , "It kills harmful creatures" , "From poison" , "What eliminates bugs" .)