どうぞ気になさらずに。

English Translation

Don’t worry about it.

is this an incomplete
/ abbreviated sentence? if so, for what?

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I am not really familiar with this, but it looks like it is a form of the honorific 為さる verb.

You may have encountered 「気にしないで」this looks like a more formal/humble version of the same sentence.
about 〜ずに, maybe the following link will help:
https://jlptsensei.com/learn-japanese-grammar/ずに-zuni-meaning/

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@ericaw
I took a look at the webpage (JLPT Sensei) that mike-lima recommended. The explanation is correct and easy for you to understand the function of (せ)ずに = without doing. It’s a good reference, indeed.

The full sentence version is:

どうぞお気になさらずにお過ごしください。= Please enjoy spending your precious time without worrying about such an ignorable person like me.

過ごす = to lead your life

And お過ごしください is so obvious and wordy that we often drop it.

お気になさる is a honorific version of 気にする. We usually put お when it is followed by なさる.

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