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If you talk around something, you don't mention it directly. You don't use more words than are needed; it's not the same as being "wordy."
You talk around something that you feel is embarrassing, that you think someone else might be embarrassed about, or just a topic we don't usually talk about in polite company. It could be anything from menopause to suicide to bowel movements, or your desire to be chosen to head a project but you don't want to ask directly.
Your example doesn't make much sense to me, but it's not an example of talking around something.
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