The default WordPress “Categories Widget” will show all categories. This can not be customized. You’ll need to install a custom plugin if you want to show only specific categories.
You might want to look into this plugin: https://wordpress.org/plugins/pro-categories-widget/
Alternatively, you can add a text widget and manually link out to the categories you want to show up.
Treat your category pages like blog posts – they need content, and to deliver value beyond “here’s a list”. A couple paragraphs is best – describe why you’ve set out the category for your visitors, and how the blog posts are related to the category. What ties them together?
If you can’t write out at least 2-3 paragraphs reasoning why you’ve pointed out the category for your visitors, we recommend you don’t call out the categories specifically.