Don't allow to get fed up (6)

I believe the answer is:

starve

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'up' is the definition.
Although both the answer and definition are verbs in their base form, I can't understand how they can define each other.

I don't understand the remainder of the clue.

Can you help me to learn more?

(Other definitions for starve that I've seen before include "Eat nothing" , "Go without food" , "Suffer from food deprivation" , "Fail to take in" , "Vaster (anag)" .)

I've seen this clue in the Evening Standard.

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