Annoy ancient matriarch carrying apple pal refused (5)
I believe the answer is:
peeve
'annoy' is the definition.
(I know that peeve is a kind of annoy)
'ancient matriarch carrying apple pal refused' is the wordplay.
'ancient matriarch' becomes 'eve' (I can't explain this - if you can you should believe this answer much more).
'carrying' is a charade indicator (letters next to each other) (in a down clue, the bottom letters carry others).
'refused' is a deletion indicator.
'apple' with 'pal' taken away is 'pe'.
'eve' put after 'pe' is 'PEEVE'.
Can you help me to learn more?
(Other definitions for peeve that I've seen before include "Cause annoyance" , "Cause to be annoyed, irritated, or resentful" , "Vex" , "grouse" , "cause irritation" .)