For this example we take code of simple plugin EntityLink. It doesn't actually it could be any plugin

use Drupal\views\Plugin\views\field\EntityLink;
use Drupal\views\ResultRow;

/**
 * Field handler to present a link to approval.
 *
 * @ingroup views_field_handlers
 *
 * @ViewsField("approve_link")
 */
class ApproveLInk extends EntityLink {

  /**
   * {@inheritdoc}
   */
  protected function getEntityLinkTemplate() {
    return 'approve';
  }

  /**
   * {@inheritdoc}
   */
  protected function renderLink(ResultRow $row) {
    $this->options['alter']['query'] = $this->getDestinationArray();
    return parent::renderLink($row);
  }

  /**
   * {@inheritdoc}
   */
  protected function getDefaultLabel() {
    return $this->t('Approve');
  }

}

We don't call constructor for our plugin. In order to pass our service. Let's take current_user service.

First we need implement 

class ApproveLInk extends EntityLink implements ContainerFactoryPluginInterface {

Add create method, where we pass our service

/**
   * @param \Symfony\Component\DependencyInjection\ContainerInterface $container
   * @param array $configuration
   * @param string $plugin_id
   * @param mixed $plugin_definition
   *
   * @return static
   */
  public static function create(ContainerInterface $container, array $configuration, $plugin_id, $plugin_definition) {
    return new static(
      $configuration,
      $plugin_id,
      $plugin_definition,
      $container->get('current_user')
    );
  }

After that add parameters to our constructor

/**
   * @var AccountInterface $account
   */
  protected $account;

  /**
   * @param array $configuration
   * @param string $plugin_id
   * @param mixed $plugin_definition
   * @param \Drupal\Core\Session\AccountInterface $account
   */
  public function __construct(array $configuration, $plugin_id, $plugin_definition, AccountInterface $account) {
    parent::__construct($configuration, $plugin_id, $plugin_definition);
    $this->account = $account;
  }