"Give me one half of that" Vs. "give me half of that"

It is perfectly acceptable to say "give me half of that". In English, "half" in understood on its own to mean "one of two equal parts of something".

To put it another way:

• It would make no sense to say "give me no halves of that". You would just say "give me none of that".
• It would make no sense to say "give me two halves of that". You would just say "give me all of that".

Saying "give me one half of that" is redundant. It's equivalent to saying "give me one of one of those two equal parts of that."

Comment posted as answer - as requested

In idiomatic usage, you would seldom say 'one half of this'. You might say 'give me one half-pound pack of sugar' but the hyphen shows that it is a different construct. You might say 'give me one third of that' (as opposed to 'two thirds of that'), but with halves, the alternatives are none and all. However, even with thirds, it would be more usual to say 'a third' than 'one third'. So, whoever 'corrected' you was actually misleading you.

There is no need to say "one half" there. "Give me half of that" is sufficient.

If you are discussing arithmetic, it is always correct to stipulate a numerator and a denominator. "Give me half an apple" is fine, but "what is half plus a third?" is incorrect and should be "what is one-half plus one-third?"