One possible approach:

>perl -wMstrict -le
"my $bad =
   'x >lt;amp; y >amp;lt; z &>< y lt;gt; z';
 print qq{'$bad'};
 ;;
 my $tity = qr{ (?: amp | gt | lt) ; }xms;
 ;;
 (my $fixed = $bad) =~
   s{ (?: (?: \G (?<! \A)) | &) $tity \K (?= $tity)  }
    '&'xmsg;
 print qq{'$fixed'};
"
'x &amp;gt;lt;amp; y &gt;amp;lt; z &amp;&gt;&lt; y lt;gt; z'
'x &amp;&gt;&lt;&amp; y &gt;&amp;&lt; z &amp;&gt;&lt; y lt;gt; z'
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