Prime Minister once having argument in airport (8)

I believe the answer is:

heathrow

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'airport' is the definition.
(London airport)

'prime minister once having argument' is the wordplay.
'prime minister once' becomes 'heath' (I can't explain this - if you can you should believe this answer much more).
'having' says to put letters next to each other.
'argument' becomes 'row' (to row is to argue).
'heath'+'row'='HEATHROW'

'in' is the link.

Can you help me to learn more?

(Other definitions for heathrow that I've seen before include "London airport" , "our counterpart of Charles de Gaulle" , "site for airport" , "London's busiest airport" , "lots of movement through the air here" .)

I've seen this clue in The Times.

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