Such city may file in the same proceeding a supplementary petition, praying the court that an assessment be made for the purpose of raising an amount necessary to pay the compensation and damages which may [be] or shall have been awarded for the property taken or damaged, with costs of the proceedings, or for such part thereof as the ordinance shall provide. The said court shall thereupon appoint three competent persons as commissioners to make such assessment, or if there be a board of eminent domain commissioners of such city, appointed under the provisions of this chapter, said proceeding for assessment shall be referred to said board. Said commissioners shall include in such assessment the compensation and damages which may [be] or shall have been awarded for the property taken or damaged, with all costs and expenses of the proceedings incurred to the time of their appointment, or to the time when said proceeding was referred to them, together with the probable further costs and expenses of the proceedings, including therein the estimated costs of making and collecting such assessment.