To get rid of it from the taskbar entirely, a better solution is to right-click the Weather icon and go to News and interests - Turn off.
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In the Windows Insider development rings, the company has been testing out a new News and Interests widget. It’s a button that lives on the right side of the taskbar. When launched it shows the weather forecast for your area, news headlines, finance, traffic, and other information. And the widget isn’t just for dev channels. It’s coming to older versions of Windows like version 2004 May 2020 Update and version 20H2 October 2020 Update in future updates
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