Neat and refined but rather slow (7)
I believe the answer is:
andante
'rather slow' is the definition.
(musical direction)
'neat and refined' is the wordplay.
'refined' is an anagram indicator.
'neat'+'and'='neatand'
'neatand' with letters rearranged gives 'ANDANTE'.
'but' is the link.
(Other definitions for andante that I've seen before include "Fairly slow (mus.)" , "quite slow" , "Moderately slow musical passage" , "Noted instruction" , "moving unhurriedly" .)