Former PM's Gulf broadcast (4)

I believe the answer is:

pitt

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'former pm's' is the definition.
(I've seen this before)

'gulf broadcast' is the wordplay.
'gulf' becomes 'pit'.
'broadcast' becomes 't' (I can't justify this - if you can you should believe this answer much more).
'pit'+'t'='PITT'

Can you help me to learn more?

(Other definitions for pitt that I've seen before include "old politician" , "Elder and Younger PMs" , "William --, early PM" , "Former British PM" , "One of two Tory prime ministers" .)

I've seen this clue in The Independent.

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