The issue here is that you are expecting persistence, whereas normally your environment gets reset for each execution of perl. If you want your variable value to persist beyond a single run, you'll need to store the value in a place that persists beyond a single execution of the script; this spec is usually hit using files for storage. So maybe you'd want something like:

#!/usr/bin/perl

package subs::utils;   
    use strict;
    use warnings;
    
    BEGIN {
        require Exporter;
        # set the version for version checking
        our $VERSION     = 1.00; 
        # Inherit from Exporter to export functions and variables
        our @ISA         = qw(Exporter);
        # Functions and variables which are exported by default
        # exported only if fully qualified
        our @EXPORT_OK   = qw($henky);
    }
    
    my $filename = 'store.txt';
    our $henky;
    if (open my $fh, '<', $filename) {
        local $/;
        $henky = <$fh>;
    } else {
        $henky = "henky_init";
    }

    END {
        open my $fh, '>', $filename or die "Storage failed: $!"
        print $fh $henky;
    }      

1;  # return a true value, standard module behaviour
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Note that this may cause some weird behavior if you don't expect a truly persistent type of behavior. Also note that my file name is unqualified, and is thus sensitive to your choice of working directory. If this is a problem, you'll need to give it a full path that everyone who might use the module has read permission on.