Are you sure @array is holding nothing but a string? If so why are you using an array?

my $str = "P H A G E P H A G E P Q K R E P H A G E P W S Q E P H A G E
+ P R D L E P H A G E";

$str =~ s/\s+//g;
my @words = $str =~ /.{1,5}/g;
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Or do you mean:

my @array = ('P', 'H', 'A', 'G', 'E', 
             'P', 'H', 'A', 'G', 'E',
             'P', 'Q', 'K', 'R', 'E',
             'P', 'H', 'A', 'G', 'E',
             'P', 'W', 'S', 'Q', 'E',
             'P', 'H', 'A', 'G', 'E',
             'P', 'R', 'D', 'L', 'E',
             'P', 'H', 'A', 'G', 'E');

my @words;
while (@array) {
   push(@words, join('', splice(@array, 0, 5)));
}
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Then, for the second part,

foreach (@words) {
   next unless $_ eq 'PHAGE';
   print("$_\n");
}
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What task are you trying to accomplish? Are you simply trying to count the occurences of runs of P H A G E? Do the occurances have to happens at particular points in the string (for instance, starting at a position that is a multiple of 5)?

If you have a string (or can make it a string), you can use index to find them and count them. Remember that index returns -1 when it doesn't find the substring, and that's a true value.

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

my $string    = "PHAGEPHAGEPQKREPHAGEPWSQEPHAGEPRDLEPHAGE";
my $substring = "PHAGE";

my $count    =  0;
my $last_pos =  -1;

NAKED:
    {
    $last_pos = index( $string, $substring, $last_pos + 1 );
    last if $last_pos == -1;
    
    print "Found $substring at $last_pos\n";
    $count++;
    redo;
    }
    
print "Found $count instances of $substring\n";

__OUTPUT__
Found PHAGE at 0
Found PHAGE at 5
Found PHAGE at 15
Found PHAGE at 25
Found PHAGE at 35
Found 5 instances of PHAGE
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If you have an array and you need to find the occurences among consecutive array entries (perhaps, because the array is huge and you want to avoid a lot of copying), you can do the same sort of thing, although you have to invent your own code to find the indices. Scoot along the array and try the subarray at each position. Once it fails, move the starting position (that's $offset) and try again. Do that until you run out of array.

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

my @array    = split //, "PHAGEPHAGEPQKREPHAGEPWSQEPHAGEPRDLEPHAGE";
my @subarray = split //, "PHAGE";

my $offset   =  0;
my $count    =  0;

NAKED:
    {
    foreach my $index ( 0 .. $#subarray )
        {
        next if $array[ $offset + $index ] eq $subarray[ $index ];
        $offset += 1;
        redo NAKED;
        }
    
    print "Found @subarray at $offset\n";
    $count++;

    $offset += @subarray;
    last if $offset + @subarray > @array;
    
    redo;
    }
    
print "Found $count instances of @subarray\n";

__OUTPUT__
Found P H A G E at 0
Found P H A G E at 5
Found P H A G E at 15
Found P H A G E at 25
Found P H A G E at 35
Found 5 instances of P H A G E
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@array = qw/P H A G E P H A G E P Q K R E P H A G E P W S Q E P H A G 
+E P R D L E P H A G E/;
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By contrast, your  @array = "P H A G E ..."; approach causes @array to contain only one element, which is a string of 79 characters (40 letters separated by 39 spaces).

So if your "previous step of the program" (whatever that may be) is really producing an array of 40 elements with one letter per element, maybe the following would do what you want:

my $start = 0;
my $end = 4;
while ( $end <= $#array ) {
    my $string = join( "", @array[$start..$end];
    if ( $string eq 'PHAGE' ) {
        print "$string found in elements $start .. $end\n";
    }
    $start += 5;
    $end += 5;
}
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