The mayor of a city of the first class 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. If the mayor approves the ordinance, order, bylaw, resolution, contract, or claim, he or she shall sign it, and it shall become effective. If the mayor vetoes the ordinance, order, bylaw, resolution, contract, or any item or items of appropriations or claims, he or she shall return it to the city council stating that the measure is vetoed. The mayor may issue the veto at the meeting at which the measure passed or within seven calendar days after the meeting. If the mayor issues the veto after the meeting, the mayor shall notify the city clerk of the veto in writing. The city clerk shall notify the city council in writing of the mayor’s veto. Any ordinance, order, bylaw, resolution, award of or vote to enter into any contract, or the allowance of any claim vetoed by the mayor, may be passed over his or her veto by a vote of two-thirds of all the members elected to the city council, notwithstanding his or her veto. If the mayor neglects or refuses to sign any ordinance, order, bylaw, resolution, award of or vote to enter into any contract, or the allowance of any claim, but fails to veto the measure within the time required by this section, the measure shall become effective without his or her signature. The mayor may veto any item or items of any appropriation bill or any claims bill, and approve the remainder thereof, and the item or items so vetoed may be passed by the city council over the veto as in other cases.