High-repetition-rate, widely tunable terahertz radiation sources are in urgent demand for applications such as material spectroscopy and rapid imaging. In this paper, the factors influencing the net gain coefficient in the THz parametric process have been theoretically analyzed. The effects of pump energy density on the THz output power and tuning range have been clarified. In experimental, THz wave with a repetition rate of 4 kHz has been achieved with the tuning range of 1.05–4.96 THz by using a magnesium oxide-doped lithium niobate crystal (MgO:LN), which was pumped by a high-repetition-rate electro-optically Q-switched 1064 nm laser. The results indicated that a higher pump energy density is beneficial for overcoming the absorption losses induced by MgO:LN crystal, thereby enhancing the output power of high-frequency terahertz waves. The experimental results are in good agreement with the theoretical calculations. Furthermore, the maximum terahertz wave average power of 120 μW was obtained at 1.7 THz at pump power of 28 W, corresponding to the energy conversion efficiency of 4.3×10^–6 and a peak power of approximately 10.3 W. This THz source offers the advantages of high average power, high peak power and widely tunable output, which shows significant potential for applications in fast THz scanning imaging and material detection.