Both are fine, but I also prefer 'furnish', as in put in the main parts of your CV. In a way, 'burnish' is more like 'polish', suggesting an extra layer on top,

Both are fine, but I also prefer 'furnish', as in put in the main parts of your CV. In a way, 'burnish' is more like 'polish', suggesting an extra layer on top, rather than the fundamental parts.