An officer or employee may be suspended, demoted, laid off or removed in the manner provided by law by the mayor or other authority which has the power to appoint or elect the officer or employee. The mayor or other authority which has the power to appoint or elect the successor of an officer or employee may appoint or elect a person to act during the temporary absence, disability or suspension of such officer or employee, or, in the case of a vacancy, until a successor is appointed or elected and qualified. The council may ordain that a particular superior or subordinate or deputy of such officer or employee shall act in such cases.