You need to keep a buffer containing the lines you may need to throw away and print lines from the buffer when you know that they won't be thrown away. This stuff is tricky to get right because you have to correctly handle special cases for start up and ending conditions. Here's one way to do it:

use strict;
use warnings;

my $kPreLines = 1;
my $kPostLines = 1;
my $skipping;
my @bufferedLines;

while (defined (my $line = <DATA>) || @bufferedLines) {
    push @bufferedLines, $line if defined $line;
    
    while (@bufferedLines && $skipping) {
        shift @bufferedLines;
        --$skipping;
    }
    
    print shift @bufferedLines while @bufferedLines > $kPreLines + 1;
    next if !@bufferedLines;
    
    if ($bufferedLines[-1] =~ /XXXXX/) {
        $skipping = $kPostLines;
        @bufferedLines = ();
        next;
    }
    
    next if defined $line;
    
    print @bufferedLines;
    last;
}


__DATA__
QQQQQ
PPPPP
XXXXX is my name
YYYYY
KKKKK
UUUUU
BBBBB
CCCCCC
XXXXX is what I play
KKKKK
NNNNN
[download]

Prints:

QQQQQ
KKKKK
UUUUU
BBBBB
NNNNN
[download]