The mayor shall have the power to approve or veto any ordinance passed by the city council, and to approve or veto any order, bylaw, resolution, award of, or vote to enter into any contract, or the allowance of any claim; provided, any ordinance, order, bylaw, resolution, award, or vote to enter into any contract, or the allowance of any claim vetoed by the mayor may be passed over such veto by a vote of two thirds of all the members elected to the council, notwithstanding such veto. If the mayor neglects or refuses to sign any ordinance, order, bylaw, resolution, award, or vote to enter into any contract, or the allowance of any claim, and returns the same with objections in writing to the next regular meeting of the council, the same shall become a law without the mayor's signature. The mayor may veto any item or items of any appropriation bill, and approve the remainder thereof. An item or items so vetoed may be passed by the council over the veto as in other cases.