Depends on the dialect of English I guess. It sounds natural in my dialect (I'm a native English speaker).
I agree with you. Over here, "cut off" doesn't sound wrong, it just sounds precise and methodical, "cut a piece" just sounds casual and common.
Just to tag onto this.
To me it depends what's being cut. You could say "cut me a piece of cake", without the 'off'. But you'd say "cut me off a piece of sellotape" or whatever. Just as an example.
-Отрезать- [typo] is when you cut something completely off of something, for example отрезать голову, "to cut off the head."
Нарезать is to systematically cut up or cut apart something, usually into slices. For example, нарезать помидор, "to cut up a tomato into slices."
Your sentence implies that the piece is being cut off of your own body, which is not what the Russian sentence says.
That would work, along with "cut a small/little piece off for me, please"
You don't need the word "off" here. It's implied. It's common to "cut me a piece of cake, please."
You may not need "off" but it still works. If you had a job as a translator you make it understandable and natural for the non Russian speaker. It doesnt change the meaning one bit.
I agree with that. The only contentious part is that отрежь is the second-person imperative tense of "to cut off." It's specifying in Russian to cut a piece off of something.
If the verb was "to cut," then in Russian the sentence would've been,
Порежь мне маленький кусок, пожалуйста.
Suppose that I am writing a beautiful story about being a pie. I(as the pie) say "cut a piece of me". How would I say that ? Отрежь меня маленький (...) ? What if I wanted be completly cut ? "Cut me in 8 parts". Thanks ^.^