I think you are overcomplicating a simple task: infact i suppose (at first glance) that your double use of the diamond <> operator is the cause of your output (PS infact every call of <> consume one line of the filehandle).

Why do not simply (untested):

while (<>){
    if (/^Relay access denied/){
       foreach (<>){
          print "$2\n" if $_ =~ /(\d+)\s+(\S+)/;
       }
    }
}
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L*