Helicity dependent photocurrent (HDPC) of the surface states in a high-quality topological insulator (Bi$_{0.7}$Sb$_{0.3}$)$_2$Te$_3$ nanoplate grown by chemical vapor deposition (CVD) is investigated. By investigating the angle dependent HDPC, it is found that the HDPC is mainly contributed by the circular photogalvanic effect (CPGE) current when the incident plane is perpendicular to the connection of the two contacts, whereas the circular photon drag effect (CPDE) dominates the HDPC when the incident plane is parallel to the connection of the two contacts. In addition, the CPGE of the (Bi$_{0.7}$Sb$_{0.3}$)$_2$Te$_3$ nanoplate is regulated by temperature, light power, excitation wavelength, the source-drain and ionic liquid top-gate voltages, and the regulation mechanisms are discussed. It is demonstrated that (Bi$_{0.7}$Sb$_{0.3}$)$_2$Te$_3$ nanoplates may provide a good platform for novel opto-spintronics devices.