Subject to the limitations on indebtedness and taxation applying to any county, city, town or village nothing in this constitution contained shall prevent a county, city or town from making such provision for the aid, care and support of the needy as may be authorized by law, nor prevent any such county, city or town from providing for the care, support, maintenance and secular education of inmates of orphan asylums, homes for dependent children or correctional institutions and of children placed in family homes by authorized agencies, whether under public or private control, or from providing health and welfare services for all children, nor shall anything in this constitution contained prevent a county, city, town or village from increasing the pension benefits payable to retired members of a police department or fire department or to widows, dependent children or dependent parents of members or retired members of a police department or fire department; or prevent the city of New York from increasing the pension benefits payable to widows, dependent children or dependent parents of members or retired members of the relief and pension fund of the department of street cleaning of the city of New York. Payments by counties, cities or towns to charitable, eleemosynary, correctional and reformatory institutions and agencies, wholly or partly under private control, for care, support and maintenance, may be authorized, but shall not be required, by the legislature. No such payments shall be made for any person cared for by any such institution or agency, nor for a child placed in a family home, who is not received and retained therein pursuant to rules established by the state board of social welfare or other state department having the power of inspection thereof.