That blank spot will be pretty small on some phones, but you can definitely add something in there.
1. Create a new Hook Element, and set it to the
generate_inside_mobile_header hook: https://docs.generatepress.com/article/hooks-element-overview/
2. In the hook content, add something like this:
<a href="URL TO YOUR HOME PAGE">Your text in here</a>
Then set your conditions in the Display Rules and it should display 🙂