A county board of commissioners may enter into an agreement or agreements for a period not exceeding 1 year with any agency, institution, or hospital, or agencies, institutions, or hospitals that have been and are for the current year licensed by the department of licensing and regulatory affairs to receive aid, care for, support, maintain, treat, cure, or relieve in or by the agency, institution, or hospital, any poor, sick, distressed, abandoned, or needy children or youth, or children or youth with special health care needs, residing within the county who may be referred to the agency, institution, or hospital by a judge of the family division of circuit court for the county in accordance with the provisions of this act, whether the aid, care, support, maintenance, treatment, cure, or relief is furnished wholly or in part by the agency, institution, or hospital. The proper charges under the contract or contracts shall be audited and paid from time to time by the board of auditors, or by the county board of commissioners of the county in counties not having a board of auditors. A county board of commissioners, before entering into a contract under this section shall fix the maximum amount to be expended for the purposes described in this section during any 1 year, which shall be raised, levied, and collected as part of the general expense of the county.