Before an ordinance passed by Council takes effect, it shall be deposited in the office of the City Clerk. If the Mayor approves the ordinance he or she shall sign the same, but if he or she does not approve the ordinance, it shall be returned to Council, with any objection thereto, in writing, at the next regular meeting of Council occurring not less than five days after the passage thereof. The disapproval or veto may extend to any one or more items or appropriations contained in an ordinance making an appropriation, or to the entire ordinance, and if the veto only extends to a part of the ordinance, the remainder thereof shall take effect and be in force. However, if the Mayor fails to return an ordinance, with his or her objections thereto, by the time of the next regular Council meeting, he or she shall be deemed to have approved the ordinance, and the same shall take effect accordingly.