Background with logo and text infant of the logo

I am not sure I got your question, but: do you mean this?

I had to fix a couple of things (see comments).

I have put all the background image in the TikZ node (but you can play with that) and made the node align multiple lines centering them)

documentclass[10pt]{exam}
    usepackage{background}
    newsaveboxmybox
    saveboxmybox{%
        tikz[opacity=0.1]node[align=center]{%
            includegraphics[width=1cm]{example-image-a}[-8pt]
        {tinytextbf{xxxx}}};
    }
    backgroundsetup{angle=0,contents={useboxmybox}}
begin{document}
    My document
end{document}

But notice that the package gives you a warning in the TeXLive 2020 release:

Package everypage Warning: Functionality similar to this package has recently
(everypage)                been implemented in LaTeX. This package is now in
(everypage)                legacy status.
(everypage)                Please, don't use it in new documents and packages.


Package everypage Warning: You appear to be running a version of LaTeX
(everypage)                providing the new functionality.
(everypage)                Doing the best to deliver the original `everypage`
(everypage)                interface on top of it. Strict equivalence is
(everypage)                not possible, breakage may occur.
(everypage)                If truly needed, Use `everypage-1x` to force the
(everypage)                loading of an older code base.