Tunes, also Greek goddesses, have delusions of grandeur (4,3,6)
I believe the answer is:
airs and graces
'delusions of grandeur' is the definition.
I don't know anything about this answer so I cannot judge whether this works.
'tunes also greek goddesses' is the wordplay.
'tunes' becomes 'airs' (air can mean a piece of music).
'also' becomes 'and' (both can mean in addition).
'greek goddesses' becomes 'graces' (I've seen this in another clue).
'airs'+'and'+'graces'='AIRS AND GRACES'
'have' acts as a link.
Can you help me to learn more?
(Other definitions for airs and graces that I've seen before include "Affectation of social superiority" , "Affectedly refined manners" , "Affectations of a 'lady', perhaps" , "Affected behaviour" , "affected attitude" .)