Here's a slightly different approach that runs a while loop checking that end of file hasn't been reached rather than reading each line directly. Inside the loop a line is pushed onto the buffer then tested for the "reject" condition. If it is a reject then the next line is also added to the buffer then the whole buffer is discarded so as to reject all three lines. If the most recent line was not a reject then the previous line is shifted off the buffer and printed.

use strict;
use warnings;

open my $inFH, q{<}, \ <<EOD or die $!;
XXXXX - additional test 1
XXXXX - additional test 1
ZZZZZ
HHHHH - additional test 1 output expected
PPPPP
XXXXX is my name 1
XXXXX is my name 2
YYYYY
KKKKK
UUUUU
BBBBB
CCCCCC
XXXXX is what I play
KKKKK
NNNNN
ZZZZZ - additional test 2b
XXXXX - additional test 2a
QQQQQ - additional test 3
RRRRR - additional test 4
GGGGG
XXXXX - additional test 5
XXXXX - additional test 5
EOD

my @buffer;
while ( not eof $inFH )
{
    push @buffer, scalar <$inFH>;
    if ( $buffer[ -1 ] =~ m{XXXXX} )
    {
        last if eof $inFH;
        push @buffer, scalar <$inFH>;
        push @buffer, scalar <$inFH>
           while $buffer[ -1 ] =~ m{XXXXX} and not eof $inFH;
        @buffer = ();
    }
    else
    {
        print shift @buffer
           unless scalar @buffer == 1 and not eof $inFH;
    }
}

close $inFH or die $!;
[download]

The output.

HHHHH - additional test 1 output expected
KKKKK
UUUUU
BBBBB
NNNNN
RRRRR - additional test 4
[download]

I hope this is of interest.

Update: Added the output.

Update: 2: Inserted push @buffer, scalar <$inFH> while ... line and added some paired "XXXXX" lines to address the point raised by mr_mischief. The output is unchanged.