Keen to have opera star sent up (4)
I believe the answer is:
avid
'keen' is the definition.
(I've seen this before)
'opera star sent up' is the wordplay.
'opera star' becomes 'diva' (diva can mean a female opera singer).
'sent up' says the letters should be written backwards.
'diva' written backwards gives 'AVID'.
'to have' acts as a link.
(Other definitions for avid that I've seen before include "In operation" , "Keenly interested" , "Very keen, desirous" , "Zealous, enthusiastic" , "deeply enthusiastic" .)