If your bibliography items are very short, an icon placed below the label number may clash with the label number of the next item. So I suggest to place the icon to the left of the label, together with the code from Audrey's answer to Creating bibliographies with biblatex and moderncv for better moderncv compatibility of the bibliography environment there is ample space there.
The icon can be taken from fontawesome5 and can be positioned with \raisebox (that bit of code is inspired by Werner's answer to How to add a symbol next to an enumerated item?) and a sprinkle of \hspace.
There are many possible ways to tell biblatex which entries are supposed to be shown with an icon. The biggest question is whether you want to tell biblatex about the icon in the .bib file or in the .tex file.
Set icon from the .bib file: options
Here I used entry options from the .bib entry, but the same principle could be applied with categories (see below).
\documentclass[british]{moderncv}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{babel}
\usepackage{csquotes}
\usepackage[style=numeric, backend=biber]{biblatex}
\usepackage{fontawesome5}
\newcommand*{\printmedalicon}{%
\raisebox{-1pt}[0pt][0pt]{\faIcon{medal}\hspace{1em}}}
\newtoggle{bbx:showicon}
\DeclareEntryOption[boolean]{showicon}{\settoggle{bbx:showicon}{#1}}
\newbibmacro*{showmedalicon}{%
\iftoggle{bbx:showicon}
{\printmedalicon}
{}}
\defbibenvironment{bibliography}
{\list
{\usebibmacro{showmedalicon}%
\printtext[labelnumberwidth]{% label format from numeric.bbx
\printfield{labelprefix}%
\printfield{labelnumber}}}
{\setlength{\topsep}{0pt}% layout parameters from moderncvstyleclassic.sty
\setlength{\labelwidth}{\hintscolumnwidth}%
\setlength{\labelsep}{\separatorcolumnwidth}%
\leftmargin\labelwidth%
\advance\leftmargin\labelsep}%
\sloppy\clubpenalty4000\widowpenalty4000}
{\endlist}
{\item}
\moderncvstyle{classic}
\moderncvcolor{burgundy}
\firstname{Jane}
\familyname{Doe}
\begin{filecontents}{\jobname.bib}
@book{appleby,
author = {Humphrey Appleby},
title = {On the Importance of the Civil Service},
date = {1980},
options = {showicon},
}
@book{elk,
author = {Anne Elk},
title = {A Theory on Brontosauruses},
date = {1972},
options = {showicon},
}
\end{filecontents}
\addbibresource{\jobname.bib}
\addbibresource{biblatex-examples.bib}
\begin{document}
\nocite{sigfridsson,aksin,appleby,elk}
\printbibliography
\end{document}

Set icon from the .tex file: Categories
\documentclass[british]{moderncv}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{babel}
\usepackage{csquotes}
\usepackage[style=numeric, backend=biber]{biblatex}
\usepackage{fontawesome5}
\newcommand*{\printmedalicon}{%
\raisebox{-1pt}[0pt][0pt]{\faIcon{medal}\hspace{1em}}}
\DeclareBibliographyCategory{showicon}
\newbibmacro*{showmedalicon}{%
\ifcategory{showicon}
{\printmedalicon}
{}}
\defbibenvironment{bibliography}
{\list
{\usebibmacro{showmedalicon}%
\printtext[labelnumberwidth]{% label format from numeric.bbx
\printfield{labelprefix}%
\printfield{labelnumber}}}
{\setlength{\topsep}{0pt}% layout parameters from moderncvstyleclassic.sty
\setlength{\labelwidth}{\hintscolumnwidth}%
\setlength{\labelsep}{\separatorcolumnwidth}%
\leftmargin\labelwidth%
\advance\leftmargin\labelsep}%
\sloppy\clubpenalty4000\widowpenalty4000}
{\endlist}
{\item}
\moderncvstyle{classic}
\moderncvcolor{burgundy}
\firstname{Jane}
\familyname{Doe}
\addtocategory{showicon}{sigfridsson,nussbaum}
\addbibresource{biblatex-examples.bib}
\begin{document}
\nocite{sigfridsson,aksin,worman,nussbaum}
\printbibliography
\end{document}
