It turns out that it wasn't obvious what you wanted to do here after all. :-)

The problem (which you haven't spelled out) is that the here doc interpolates when you set $a so your initial $y is never changed. One solution is not to interpolate but to use a placeholder scalar (which could just be "$y" again) and then eval the here document.

#!/usr/bin/env perl 
use strict;
use warnings;

my ($y, $out);

$y = 'baby';
my $hd = <<'EOT';
Hello $y!
EOT

eval "\$out = qq{$hd}";
print $out;
$y = 'world';
eval "\$out = qq{$hd}";
print $out;
[download]

But that's a bit crude. Far better to use a proper templating system.