This seems the easiest way to me…

use 5.010;
use strict;
use warnings;

{
    package Test;
    use Moose;
    my $prepend = '/some_string/';
    has _appended => (is => 'rw', init_arg => 'appended');
    sub appended { $prepend . shift->_appended(@_) }
}

my $t = Test->new(appended => "this_is_appended_string");
say $t->appended;
$t->appended('another_string');
say $t->appended;
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