/** @var dbal */

	protected $db;

Invalid class dbal, should be \phpbb\db\driver\driver

* @param mixed $cache

Parameter $cache of the contructor should be \phpbb\cache\service in the doc block as it is type casted anyway

Parameter $include_path of method add() is unused, Should it be used? it's used in the remove() method

remove():
The fetched field does not match any selected column.
Also the variable module_name is unused later, therefor we could just remove the query from the else all together:

			else

			{

				$module = (int) $module;

				$sql = 'SELECT module_langname

					FROM ' . $this->modules_table . "

					WHERE module_id = $module

						AND module_class = '" . $this->db->sql_escape($class) . "'

						$parent_sql";

				$result = $this->db->sql_query($sql);

				$module_name = $this->db->sql_fetchfield('module_id');

				$this->db->sql_freeresult($result);

				$module_ids[] = $module;

			}

EXreaction can you have a look?