No suit shall be brought upon any claim for money or damages against the City of Eureka until a demand for the same has been presented as herein provided, and rejected in whole or in part. If rejected in part, suit may be brought to recover the whole. Nor shall suit be brought against said city upon any such claim or demand, if the same shall be in whole approved and audited as provided herein; provided, that nothing herein contained shall be construed so as to deprive the holder of any demand of his right to resort to a writ of mandate, or other proceedings against the said Council, or any Board, Commission or Committee or officer of said city, to compel it, or him, to act upon such demand or claim, or to pay the same when so audited.