Carthaginian holds educated woman to be strait-laced (9)
I believe the answer is:
puritanic
'be strait-laced' is the definition.
(I've seen this before)
'carthaginian holds educated woman' is the wordplay.
'carthaginian' becomes 'punic' ('Punic' can be a synonym of 'carthaginian').
'holds' indicates putting letters inside.
'educated woman' becomes 'rita' (ref to play and film Educating Rita).
'punic' placed around 'rita' is 'PURITANIC'.
'to' is the link.
(Other definitions for puritanic that I've seen before include "Very strict" , "Strait-laced" .)