Hello,
We apologize for the issue you are facing with changing the Focus keyword on your homepage. To set a Focus keyword for your homepage, please ensure that you have selected a specific page as your homepage under WordPress Dashboard > Settings > Reading > Your homepage displays. If you haven’t set a specific page, you won’t be able to add focus keywords to it using Rank Math.
Once you have confirmed that a specific page is set as your homepage, you can edit the SEO Title and Description directly from the Rank Math Metabox of that page. You can also access the settings by going to Rank Math > Titles & Meta > Homepage if your homepage is not set to a static page.

Regarding the issue with Rank Math being unable to read your mobile site, this could be due to various reasons such as JavaScript errors or plugin conflicts. We recommend checking for any JavaScript-related errors in your browser’s console or reviewing our guide on plugin conflicts to troubleshoot this further.
If you continue to experience issues, please reply to this ticket with your site URL, and a human agent from the Rank Math Support team will assist you further. Thank you for raising this concern!
Sources:
How to Fix the "No Keywords Found in the Page Title" Test in Rank Math » Rank Math
Why Your SEO Title, Description & Focus Keyword Aren’t Being Saved » Rank Math
Does not function! Sorry!
Hello,
If you’re using the latest post on your homepage, then you won’t need to set the focus keyword.

You’ll just need to set your meta tags on your homepage in Rank Math > Titles & Meta > Homepage:
https://rankmath.com/kb/homepage-meta-title-and-description/#edit-your-titles-meta-settings
Regarding the mobile, Rank Math uses an external script (Thum.io) for that snapshot, so we cannot do anything from our side.
That screenshot is only updated when the service’s cache runs out, which usually happens after a week or so. However, please note that this will not affect your live site’s mobile screenshot as search engines test your current site’s state
Looking forward to helping you.
Hello,
Since we did not hear back from you for 15 days, we are assuming that you found the solution. We are closing this support ticket.
If you still need assistance or any other help, please feel free to open a new support ticket, and we will be more than happy to assist.
Thank you.