Place of teaching: here CID holds mole (beaten up) (6,6)
I believe the answer is:
middle school
'place of teaching here' is the definition.
(where children are taught)
'cid holds mole beaten up' is the wordplay.
'beaten up' indicates anagramming the letters.
'cid'+'holds'+'mole'='cidholdsmole'
'cidholdsmole' anagrammed gives 'MIDDLE SCHOOL'.
(Other definitions for middle school that I've seen before include "Level of education" , "Its pupils are aged from 9 to 13" .)