Beamer: handout/article mode - produce multiple copies of a frame with distinct overlay numbers

You do need to tell beamer what you want to appear in each frame of the handout version. You can use \onslide specifications to do this so that you do not need to modify the existing overlay specifications at all. (I'm not sure this is what you want - probably not.)

The following code modifies your MWE so that, in presentation mode, there would actually be 10 slides within the frame. These are then placed on slides 1 or 2 of the handout. This is a waste of code: if you only want to show slide 5 and slide 10, you do not need two \againframe but only one. But, anyway, with the two:

\documentclass[handout]{beamer}

\begin{document}

  \begin{frame}<1-| handout:0>[label=foo]{My only frame}
    \begin{enumerate}[<+->]% or whatever you like for presentation mode
      \onslide<1-| handout:1-2>
      \item Item \arabic{enumi}
      \item Item \arabic{enumi}
      \item Item \arabic{enumi}
      \item Item \arabic{enumi}
      \item Item \arabic{enumi}
      \onslide<1-| handout:2>
      \item Item \arabic{enumi}
      \item Item \arabic{enumi}
      \item Item \arabic{enumi}
      \item Item \arabic{enumi}
      \item Item \arabic{enumi}
    \end{enumerate}
  \end{frame}

  \againframe<0| handout:1>{foo}
  \againframe<0| handout:2>{foo}
\end{document}

More efficiently with only one repeat:

\documentclass[handout]{beamer}

\begin{document}

  \begin{frame}<1-| handout:1>[label=foo]{My only frame}
    \begin{enumerate}[<+->]% or whatever you like for presentation mode
      \onslide<1-| handout:1-2>
      \item Item \arabic{enumi}
      \item Item \arabic{enumi}
      \item Item \arabic{enumi}
      \item Item \arabic{enumi}
      \item Item \arabic{enumi}
      \onslide<1-| handout:2>
      \item Item \arabic{enumi}
      \item Item \arabic{enumi}
      \item Item \arabic{enumi}
      \item Item \arabic{enumi}
      \item Item \arabic{enumi}
    \end{enumerate}
  \end{frame}

  \againframe<0| handout:2>{foo}
\end{document}

I'm not sure how this could apply to article mode since then there are no frames or slides.