I think what probably should happen here is that a user's personalized results should have an effect on the widgets that appear.
If someone blocks Wikipedia from their results, Kagi should automatically no longer display the widget on the side that shows the Wiki for that query.
Problem solved.
Kagi allows such a high degree of customization that personal choices like "no Wikipedia" or "no mainstream media" or "no Facebook" should fully reflect in their search experience, provided Kagi is not superimposing those sources elsewhere like as in the widgets.
In the case of Wikipedia of course as @Jop has said, you can already disable the widget.
