Documentation for MooseX::Clone is pretty clear about that, add 'NoClone' trait to attributes you don't want to clone:

use 5.010;
use strict;
use warnings;

{

    package Foo;
    use Moose;
    with 'MooseX::Clone';

    has bar => ( is => 'rw', default => 42, );
    has baz => ( is => 'rw', default => 42, traits => [qw(NoClone)], )
+;
}

my $foo = Foo->new( bar => 33, baz => 33 );
my $clone = $foo->clone;

say "Bar: ", $clone->bar;
say "Baz: ", $clone->baz;
__END__
Bar: 33
Baz: 42
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