Yes, you can remove items and move other ones.  That's the beauty of themes - you can edit the code to make the header (or any other element) appear exactly the way you want it to.  You just need the skills to edit code.  And that skill should be pre...

Here's a liquid-only way to do it - https://jadepuma.com/blogs/blog/diy-cart-promotions.  The code in the article should help you get started.