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Context: ask someone to eat the pizza fully and leave the residue of it.

Please finish the pizza fully.

Please put/keep half of the pizza for me.

Please leave the pizza a little bit.

Please check them, especially those verbs.

Is "put" suitable here?

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kumenglishContext: ask someone to eat the pizza fully and leave the residue of it.

It is not possible to eat something fully and also have something that remains of it (residue), so this is contradictory. Maybe you meant "or" instead of "and"? That would be written like the following:

The context is to encourage someone to eat the whole pizza or to ask them to save some for you.

(We don't use the word 'residue' for the idea of left-over food.)

kumenglishPlease finish the pizza fully.

Or,

Please finish the pizza.
Please finish the whole pizza.
Please eat the whole pizza.
Feel free to eat the whole pizza.
Don't hesitate to eat the whole pizza.

kumenglishPlease put/keep half of the pizza for me.

'keep' is OK there. 'put' is not OK there.

'leave' and 'save' are also OK.

Please leave half of the pizza for me.
Please save half of the pizza for me.

kumenglishPlease leave the pizza a little bit.

Please leave a little pizza (for me).
Please leave a little bit of pizza (for me).
Please leave some pizza (for me).

'save' is also OK instead of 'leave' in these three above.

kumenglishIs "put" suitable here?

Answered above.

CJ