Untested:

my $buffer = "";
while (<FH>) {
    next unless /\S/;   # Skip blank lines.
    if (/^\D/) {        # A bar is a non-digit as well.
        chomp $buffer;
        $buffer .= $_;
        next;
    }
    print $buffer;
    $buffer = $_;
}
print $buffer;
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Another way:

my $nl = "";
while (<FH>) {
    next unless /\S/;
    $nl = chomp;
    print $nl unless /^\d/ || $. == 1;
    print;
}
print $nl;  # Prints newline if we saw at least one non-blank file.
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