a. The mayor, not later than the fifth day after the council has acted upon the budget or capital program submitted with the executive budget, may disapprove any increase or addition to the budget, any unit of appropriation, or any change in any term or condition of the budget. The mayor, by such date, may also disapprove any item or term or condition included in such budget pursuant to the provisions of section two hundred forty-three of this chapter. The mayor shall return the budget by that date to the council, setting forth objections in writing.
b. The council, by a two-thirds vote of all the council members, may override any disapproval by the mayor pursuant to subdivision a of this section; provided, however, that if no such action by the council is taken within ten days of such disapproval, the expense budget to which such disapprovals relate shall be deemed adopted as modified by the disapprovals by the mayor.