Any town board may expend town road and bridge funds upon a legally established cartway the same as on town roads if, in the judgment of the board the public interests require it; provided, that where any town board has refused to allocate funds for the upkeep of a cartway, then, upon the petition of ten taxpayers of the town, the town board shall present for the approval of the voters, after due notice, at the annual town meeting the petition for allocation of funds, and at the town meeting the electors of the town shall allow or reject the petition. If the majority of those voting approve the petition for allocation of funds, the town board shall expend road and bridge funds on the cartway.