Adding a join in a Views field handler class constructor?
Yes, but you need to have already defined that relationship in the view. Per the docs for views_handler_field::add_additional_fields($fields):


  @param $fields
  An array of fields. The key is an identifier
  used to later find the field alias used. The value is either a
  string in which case it's assumed to be a field on this handler's
  table; or it's an array in the form of
  @code array('table'
  => $tablename, 'field' => $fieldname) @endcode


Thus, say you had a field handler for a field on the node table with a defined author relationship to the users table. You could do this:

  function construct() {
    parent::construct();
    $this-&gt;additional_fields['mail'] = array(
      'table' =&gt; 'users',
      'field' =&gt; 'mail',
    );
  }


Which would allow you to do this:

function render($values) {
  //Get the mail column from the users table.
  $this-&gt;get_value($values, 'mail');
}


For your case, you'll need to define the entity relationship to your base table before you can do this.