All contracts relating to the affairs of such city shall be in writing, signed and executed, in the name of the city, after due notice, by the officer authorized to make the same, and, in cases not otherwise directed by law or ordinance, such contracts shall be made and entered into by the mayor. No contract shall be entered into or executed directly by the city council or by a committee thereof, but some officer of the executive branch of government of such city shall be designated by ordinance to enter into and execute the same. All contracts shall be countersigned by the city controller, and filed and registered by number, date, and contents in the mayor's office, and attested copies furnished to the city controller and to the department charged with the work.