Ok, let me rewrite your code for you in a simpler way... You want to match in the "second" name if I get it right.. avoiding matching people who might be called Jackson Anything or Jackson Something or perhaps, even just jackson. I have put comments in the code itself.

my @myNames = ('Larry jackson', 'Curly jackson', 'Moe Sanders','Jackso
+n Something','jackson',);   

# our list of names, note that I added a few more

 print "Who's on the list:\n"; 

 foreach $fullname (@myNames) {             

        # $fullname contains each name in @myNames one by one

        @array = split (/\s+/,$fullname);   

        # lets split the name based on a whitespace character, 
        # in our case, a space, \s

        if (scalar @array > 1) {       

        # if the number of elements in the array @array 
        # is more than one, which it should be, then 
        # check the second element, like this

           if($array[1] =~ /jackson/i) {   

            # the "i" indicates a case-insensitive match,  
            # Jackson is same as jackson

                print "$fullname\n"; # go ahead and print 

           }

        }

 }
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