High-temperature superconducting (HTS) magnets have found wide applications in high-field settings owing to their high current capabilities. Typically, these magnets are powered by high-current power supplies via current leads, which can complicate insulation between cryogenic and room temperature environments. However, new developments in flux pumps for HTS magnets have enabled charging of kA levels of current without power supplies. By combining flux pumps with HTS persistent current operation, it is possible to achieve accurate flux modulation and eliminate the need for power supplies and current leads. In this study, we report on a novel feedback-controlled flux modulation for HTS magnets in persistent current operations. This flux modulation is based on a flux pump mechanism that generates a DC voltage across the charging superconductor by applying a current higher than its critical current. With closed-loop feedback control, our flux modulation can achieve precise injection and reduction of HTS magnet current in increments of 0.5 A. This technology can lead to stable magnetic fields in HTS magnet designs. We anticipate that this work will enable future magnets to operate in a stable persistent current mode within a closed cryogenic chamber, significantly reducing the footprint and power demand of HTS magnets and opening up new opportunities for their applications.