The case for a little more tuition (4)
I believe the answer is:
etui
'the case' is the definition.
(case for needlework equipment)
'a little more tuition' is the wordplay.
'a little' says the answer is hidden in the clue (a little bit of the clue).
'ETUI' is hidden in the letters of 'more tuition'.
'for' acts as a link.
(Other definitions for etui that I've seen before include "Small needle case e" , "Small ornamental case" , "a seamstress requires?" , "Case containing sharp objects" , "Small case for small articles" .)