Created attachment 2403 
A proposed patch.

When the summary window is closed through the window manager (such as through a key combination), it closes the window and then opens up the help window.

I believe piece of code in panel-plugin/weather-summary.c is responsible:

	if (response == GTK_RESPONSE_ACCEPT)
		gtk_widget_destroy(window);
	else
 		g_spawn_command_line_async ("exo-open --launch WebBrowser " PLUGIN_WEBSITE, NULL);

To fix this, I altered that to:

	if (response == GTK_RESPONSE_HELP)
		g_spawn_command_line_async ("exo-open --launch WebBrowser " PLUGIN_WEBSITE, NULL);
	else
 		gtk_widget_destroy(window);

which seems to fix this problem.