Whenever the city government lays out any new street or public way, or widens or otherwise alters or discontinues any street or way in a city, and decides that any persons or corporations are entitled to damage therefor, and estimates the amount thereof to each in the manner provided by law, it may apportion the damages so estimated and allowed, or such part thereof as to it seems just, upon the lots adjacent to and bounded on such street or way, other than those for which damages are allowed, in such proportions as in its opinion such lots are benefited or made more valuable by such laying out or widening, alteration or discontinuance, not exceeding in case of any lot the amount of such benefit, but the whole assessment shall not exceed the damages so allowed. Before such assessment is made, notice shall be given to all persons interested of a hearing before said city government, at a time and place specified, which notice shall be published in some newspaper in said city at least one week before said hearing.