Tap leaking at the back and Edward's annoyed (6)
I believe the answer is:
bugged
'annoyed' is the definition.
(I've seen this in another clue)
'tap leaking at the back and edward's' is the wordplay.
'tap' becomes 'bug' (synonyms).
'at the back' suggests the final letters.
'and' means one lot of letters go next to another.
'edward' becomes 'Ed' (short for Edward).
The final letter of 'leaking' is 'g'.
'bug'+'g'+'ed'='BUGGED'
(Other definitions for bugged that I've seen before include "Irritated; planted concealed microphones" , "Fitted with hidden microphone for eavesdropping" , "Fitted with illegal listening or recording device" , "Of room, fitted with secret microphones" , "Softly glowing" .)