question

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You could cancel the closing of the flyout in its Closing event handler to prevent a flyout from being hidden when clicked.

Please check the following code as a sample:

MainPage.xaml

 <AppBarButton x:Name="Reports" Icon="Page2" Label="Reports">
     <AppBarButton.Flyout>
         <MenuFlyout x:Name="ReportFlyout" Closing="ReportFlyout_Closing">
             <ToggleMenuFlyoutItem Name="TogleStuff" Click="TogleStuff_Click" Text="click me"/>
         </MenuFlyout>
     </AppBarButton.Flyout>
 </AppBarButton>

MainPage.xaml.cs

 private bool flag = false;  //The flag is used to indicate if the TogleStuff is clicked.
    
 private void TogleStuff_Click(object sender, RoutedEventArgs e)
 {
     flag = true;
     //Do other things.
 }
    
 private void ReportFlyout_Closing(FlyoutBase sender, FlyoutBaseClosingEventArgs args)
 {
     if(flag==true)
     {
         args.Cancel = true;  //If the TogleStuff is clicked, cancel the closing process.
     }
     flag = false;
 }

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