Annoyed with daughter after Henry’s middle age (7)
I believe the answer is:
enraged
'annoyed' is the definition.
(both can mean irritated or angered)
'daughter after henry's middle age' is the wordplay.
'daughter' becomes 'd' (genealogical abbreviation).
'after' says to put letters next to each other.
'middle' indicates the central letters.
The central letters of 'henry' are 'enr'.
'enr'+'age'='enrage'
'd' after 'enrage' is 'ENRAGED'.
'with' is the link.
(Other definitions for enraged that I've seen before include "Maddened" , "with all guns blazing" , "Made very angry" , "Furious" , "Infuriated, made mad" .)