Whenever it is desired to make any improvement or improvements authorized in section 14-385, where the costs of such improvement or improvements are to be assessed against the adjacent and abutting property benefited thereby, and no petition has been filed therefor in accordance with section 14-391, the city for that purpose may propose such improvement or improvements stating the specific character of the improvement or improvements thus to be made. The city shall cause to be published in the official newspaper a brief notice of such proposal stating the character of the improvement or improvements proposed thereby, and shall give additional notice to the property owners in the district or districts, or proposed district or districts, as required by section 25-520.01. If within thirty days thereafter the owners of fifty-one percent of the taxable property abutting upon the street or streets, or part or parts thereof proposed to be improved protest against such project, such work shall not be done. In the absence of such protest, the city shall be authorized to proceed with the work as proposed. The cost and expense thereof, as provided by law, may be assessed against the property within the district or districts specially benefited to the extent of such benefits as a special assessment. Where assessment against the property within the district or districts specially benefited is not made, or where the improvement or improvements are on a main thoroughfare, major traffic street, or connecting link, or made pursuant to sections 14-3,103 to 14-3,106, this section shall not apply.