Henry left Theresa outside - that's not appreciated (9)
I believe the answer is:
thankless
'not appreciated' is the definition.
'henry left theresa outside' is the wordplay.
'henry' becomes 'hank'.
'left' becomes 'l' (common abbreviation).
'theresa' becomes 'tess'.
'outside' is an insertion indicator.
'hank'+'l'='hankl'
'hankl' going inside 'tess' is 'THANKLESS'.
'that's' is the link.
(Other definitions for thankless that I've seen before include "Receiving no appreciation" , "Showing no gratitude" , "Unappreciated" .)