In the allowed motion for funds that your attorney provided, a rate was approved in addition to the total amount.
Expert and vendor rates are set by CPCS, not the courts. Any rate in excess of the vendor's assigned rate must be approved by appropriate Deputy Chief Counsel, Director of the Mental Health Litigation Unit, or Director of YAD, as described here (4th paragraph).
Section 9(i) of G.L. c. 211D authorizes CPCS to establish qualifications and rates for expert witnesses, investigators, and social service providers who are paid as vendors in accordance with the Indigent Court Costs Act, G.L. c. 261, §§ 27A-27G.