If, in case of such excess, the vote of the precinct where the error occurred would change the result as to a county office if the person appearing to be elected were deprived of so many votes, then the election shall be set aside as to that person in that precinct, and a new election ordered therein; but no person who was not a registered voter in that precinct at the time of the general election shall be allowed to vote at such special election. If the error occurs in relation to an office of a city, school district, township, or of any special district whose elections may be conducted under this chapter, the governing body of the political subdivision involved may order a new election or not, in their discretion.