There isn’t an option to display Categories + Featured topics and the latest topics list. There is an option to display Categories and Latest Topics on the home page though. Maybe this would meet your client’s requirements.
The easiest way to change your homepage options is to re-run the setup wizard by going to https://forum.yourdomain.com/wizard. Skip through the Wizard’s steps until you get to the Homepage step. You can use the dropdown menu on that step to preview the different homepage options.
If you don’t want to re-run the setup wizard, you can change the category display style by adjusting the desktop category page style site setting.
If none of the available options meet your client’s needs, it could be possible to achieve what you are looking for with a custom theme. You can find details about how to get started with that here: Developer’s guide to Discourse Themes. One thing to note about theme customizations that override Discourse templates is that they can break due to updates in the core Discourse code. If it is possible, it is best to avoid overriding the Discourse templates.