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In this paper, the design of a 4th-harmonic gyrotron with 1-kW-level output power at 1 terahertz (THz) is presented. A unique advantage of this design is that, with a well-optimized magnetic-cusp large-orbit electron gun, the designed gyrotron can operate from 1st harmonic to 4th harmonic in multiple discrete bands by varying the operating voltage. By carefully balancing the competing modes, the gyrotron can be tuned to operate at six candidate modes, including TE1,2 for fundamental harmonic, TE2,3 and TE2,4 for 2nd harmonic, TE3,5 and TE3,6 for 3rd harmonic, and TE4,8 for 4th harmonic interactions. As the main operating mode, the TE4,8 can generate a peak output power of 1.68 kW, with an efficiency of about 2.1%, and a magnetically controlled frequency tuning range of about 1.8 GHz around 1 THz. The impact of the longitudinal nonuniformity of the cavity at high-harmonic interaction was also studied. It shows a radial tolerance of several micrometers will significantly elevate the start-oscillation current and deteriorate output performance. This design is towards the development of a synthesizing THz source with ultra-wideband tuning capability ranging from the sub-millimeter-wave to 1-THz bands.