If the county mayor, when a bill of costs authenticated as provided for in § 5-9-308 is presented to the county mayor, and the county mayor’s warrant for the payment of the same demanded, conceives that the bill of costs, or any part of it, is not lawfully chargeable to the county, the county mayor may defer the issuance of the county mayor’s warrant until the county mayor has moved the court for a correction of the taxation.