Message in very odd parts of zodiac wanting flipping Everyman to irritate (5,4)
I believe the answer is:
voice mail
'message' is the definition.
'very odd parts of zodiac wanting flipping everyman to irritate' is the wordplay.
'very' becomes 'v' (abbreviation).
'odd parts of' indicates alternate letters (letters in odd-numbered positions).
'wanting' says to put letters next to each other.
'flipping everyman' becomes 'em' (I can't justify this - if you can you should give a lot more credence to this answer).
'to irritate' becomes 'ail'.
The alternate letters of 'zodiac' are 'oic'.
'v'+'oic'+'em'+'ail'='VOICE-MAIL'
'in' acts as a link.
Can you help me to learn more?
(Other definitions for voice mail that I've seen before include "Audible message left on a phone" , "messaging system" .)