I note incompleteness of ancient city for a start (5)
I believe the answer is:
intro
'for a start' is the definition.
(introduction)
'i note incompleteness of' is the wordplay.
'note' becomes 'n' (abbreviation).
'incompleteness' becomes 'tr' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'of' becomes 'o''.
'i'+'n'+'tr'+'o'='INTRO'
'ancient city' is the link.
This may not be right. Some or all of it may be part of another bit of the clue.
Can you help me to learn more?
(Other definitions for intro that I've seen before include "Early music piece" , "starter" , "Song's first few bars" , "Musical opening" , "Preamble" .)