One university offered Dawn's song (6)
I believe the answer is:
aubade
'dawn's song' is the definition.
(poem or piece of music inspired by dawn)
'one university offered' is the wordplay.
'one' becomes 'a' (a thing is one thing).
'university' becomes 'U' (abbreviation for university).
'offered' becomes 'bade' (I've seen this in another clue).
'a'+'u'+'bade'='AUBADE'
(Other definitions for aubade that I've seen before include "early composition" , "Festive occasion" , "Morning song" , "Dawn music or poem" , "piece of early music" .)