Such funds as are received from the United States and agencies thereof as grants-in-aid for the purpose of providing public assistance and social services grants shall be paid monthly by the Commissioner to each county, city or district fiscal officer as reimbursement of the federal share of such grants as have been paid by each county and city under the provisions of Subtitle II and III of this title. Within the limits of the appropriations of state funds, the Commissioner shall reimburse the entire balance of such public assistance and social services grants as have been paid by each city, county or district fiscal officer after crediting them with the reimbursement made from federal funds. Within the limits of the appropriations of state funds, the Commissioner shall reimburse monthly each city, county or district fiscal officer to the extent of sixty-two and one-half percent of such expenditures made in connection with general relief provided under § 63.2-802. Within the limits of the appropriations of state funds for the purpose, the Commissioner shall reimburse monthly each city, county or district fiscal officer to the extent of eighty percent of expenditures made for auxiliary grants pursuant to § 51.5-160. Within the limits of state funds appropriated for the purpose, the Commissioner shall reimburse to each county, city or district fiscal officer an amount not less than fifty percent or more than sixty-two and one-half percent of such expenditures, not federally reimbursable, made for the care of children placed in family homes or institutions pursuant to §§ 63.2-900 and 63.2-903. Administrative expenditures made by the localities in connection with the providing of public assistance grants, other benefits and related social services, including child welfare pursuant to § 63.2-319, shall be ascertained by the Board, and the Commissioner shall, within the limits of available federal funds and state appropriations, reimburse monthly each county, city or district fiscal officer therefor out of such federal and state funds in an amount to be determined by the Board not less than fifty percent of such administrative costs. The Commissioner also shall reimburse monthly, to the extent funds are available for such purpose, each county, city or district fiscal officer out of state and federal funds, to the extent provided in the preceding paragraph, for monthly rental payments for office space provided the local department in publicly owned buildings, for payments that are based on the cost of initial construction or purchase of a building or a reasonable amount for depreciation of such building, and for the cost of repairs and alterations to either a privately or publicly owned building. However, no monthly rental payment shall exceed a reasonable amount as determined by the Commissioner. Claims for reimbursement shall be presented by the local board to the Commissioner, and shall be itemized and verified in such manner as the Commissioner may require. Such claim shall, upon the approval of the Commissioner, be paid out of funds appropriated by the Commonwealth and funds received from the federal government for the purposes of Subtitles II and III of this title, to the treasurer or other fiscal officer of the county or city. Wherever two or more counties or cities have been combined to form a district pursuant to § 63.2-306, reimbursements by the Commissioner under this section shall be paid to the district fiscal officer or other person designated to receive such funds by the governing bodies of such counties or cities. The Commonwealth shall reimburse each county and city the full amount of public assistance grants provided for Temporary Assistance for Needy Families.