I think you want push:

#!/usr/bin/perl
use warnings;
$word = "ABCDEFGHIJKLMNO";
## Determining the number of 4-letter Fragments:
$a         = length($word);
$four      = 4;                # Line 5
$frag_Num1 = $a / $four;
$frag_Num  = int $frag_Num1;
print "Total Fragments= $frag_Num\n\n";
print "The 4-Letter Fragments are:\n\n";
$frag = 0;

while ( $word =~ /(.{4}?)/igs )    # Line 10
{
    $frag++;
    $b    = $&;
    $pos  = $-[0] + 1;
    $leng = length($b);
    print "Fragment: $&; Fragment Number: $frag of length $leng letter
+s Starting at position: $pos\n";
    print "Its length= $leng Letters;\n\n";    # Line 15
    $newlist = $leng;
    push @newarray, $newlist;
}
print "\n Newarray (should be 4 4 4): @newarray\n\n";    # Line 19
exit;

__END__

Last line of output is now...

 Newarray (should be 4 4 4): 4 4 4
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• I ran your code through perltidy (a little whitespace is a good thing).