Pushy promotion from pushy Peter (4)

I believe the answer is:

hype

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'pushy promotion' is the definition.
'hype' can be an answer for 'promotion' (hype is a kind of promotion). I'm unsure of the 'pushy' bit.

'from pushy peter' is the wordplay.
'from' indicates a hidden word.
'HYPE' is hidden in the letters of 'pushy peter'.

Can you help me to learn more?

(Other definitions for hype that I've seen before include "Extravagant promotion of a person or product" , "Overdo promotion" , "Intensive promotion or excitement" , "Intensive publicity" , "Excessive publicity" .)

I've seen this clue in The Sun.

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