The county board may, with the approval of the mayor and council or the chairperson and board of trustees, as the case may be, whenever conditions warrant, furnish, deliver, and spread gravel of a depth not exceeding three inches on certain streets in cities of the second class and villages having a population of not more than fifteen hundred inhabitants as determined by the most recent federal decennial census or the most recent revised certified count by the United States Bureau of the Census and shall charge the cost of such improvement to that portion of the Highway Allocation Fund allocated to such counties from the Highway Trust Fund under section 39-2215. No improvement of any street or streets in cities of the second class or villages having a population of not more than fifteen hundred inhabitants as determined by the most recent federal decennial census or the most recent revised certified count by the United States Bureau of the Census shall be made under the provisions of this section unless the street or streets, when graveled, will constitute one main thoroughfare through such city or village that connects with or forms a part of the county highway system of such county which has been or which shall be graveled up to the corporate limits of such city or village. Before being entitled to such county aid in graveling such thoroughfare, the same must have been properly graded by such city or village in accordance with the grade established in the construction of the county road system.