Belittle the little beauty on the cover (8)

I believe the answer is:

diminish

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'belittle' is the definition.
(I know that belittle can be written as diminish)

'little beauty on the cover' is the wordplay.
'little' becomes 'mini'.
'beauty on' becomes 'dish' (I can't justify this - if you can you should believe this answer much more).
'the cover' means one lot of letters goes inside another.
'mini' inserted within 'dish' is 'DIMINISH'.

'the' acts as a link.

Can you help me to learn more?

(Other definitions for diminish that I've seen before include "Make less with mini dish" , "sink" , "Grow smaller, shrink" , "get shorter" , "Contract" .)

I've seen this clue in the Evening Standard.

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