Badly mangled a college’s name (8)
I believe the answer is:
magdalen
'college's name' is the definition.
(as in Magdalen College, Oxford)
'badly mangled a' is the wordplay.
'badly' indicates an anagram (letters in the wrong order).
'mangled'+'a'='mangleda'
'mangleda' anagrammed gives 'MAGDALEN'.
(Other definitions for magdalen that I've seen before include "Mary ........, companion to Christ" , "Mary - Oxford college" , "Mary . . . . . . . ., follower of Christ" , "Infamous laundries" .)