A minigyrotron scheme controlled by a compact pulse magnet to excite broadband terahertz (THz) radiation is presented here. In comparison to an open-cavity circuit, the adopted prebunched backward-wave interaction circuit can expand tuning bandwidth tenfold under the control of time-varying magnetic field strength, which also significantly extends the available duration time of the pulse magnet for gyrotron operation. A quasi-optical mode convertor and a Brewster window constitute the output system to transfer the broadband radiation from the circuit into free space. A systematic gyrotron design is also presented. Driven by a low-voltage electron beam, the minigyrotron is predicted to generate radiation with 10-GHz tuning bandwidth around 0.33 THz and a maximum peak power of 2.1 kW with 6-ms pulse duration, using a TE6,2 mode interaction. Such a THz gyrotron with broad tunable bandwidth, kilowatt level power, and with the unique advantage of a compact configuration is the key to high-power THz scientific and industrial applications.