‘Newsletter’ plugin was written in special way: it defines admin menu items with virtual capability ‘exist’ which available to any user.

‘Newsletter’ applies/checks permissions before it defines admin menu items. It’s not friendly to the WordPress user capabilities system. It uses directly WordPress built-in roles: administrator or editor (if it’s allowed at the plugin settings page – “Enable access to blog editors”. Parts of menu is available for the ‘administrator’ role only:

function is_allowed() {
        return current_user_can('administrator') || $this->options['editor'] == 1 && current_user_can('editor');

    function admin_menu() {
        if (!$this->is_allowed()) return;
        add_menu_page('Newsletter', 'Newsletter', 'exist', 'newsletter_main_index', '', plugins_url('newsletter') . '/images/menu-icon.png', '30.333');

        $this->add_menu_page('index', __('Dashboard', 'newsletter'));
        $this->add_admin_page('info', __('Company info', 'newsletter'));
        if (current_user_can('administrator')) {
            $this->add_menu_page('welcome', __('Welcome', 'newsletter'));
            $this->add_menu_page('main', __('Settings and More', 'newsletter'));
            $this->add_admin_page('smtp', 'SMTP');
            $this->add_admin_page('status', __('Status', 'newsletter'));

So User Role Editor will not help you with access to this plugin for a custom role. It would be possible only after direct editing of the code above and other.