Enhanced photonic microwave generation by using a filtered optical feedback in an optically injected semiconductor laser operating at period-one (P1) dynamics is numerically demonstrated. In the simulation, the frequency tunability of the generated narrow-linewidth photonic microwave with the filtered optical feedback has been investigated. The results show that the frequency of the narrow-linewidth photonic microwave can be widely tuned by adjusting the injection parameters only or adjusting both the injection parameters and the center frequency of the filter. Moreover, the influence of the delay time, feedback strength, filter bandwidth and detuning on the linewidth, side-peak suppression and phase noise of the generated microwave have also been investigated in detail. The results show that with increasing feedback strength or delay time, evident reduction of the linewidth is observed. The side-peak suppression also increases with increasing feedback strength; however, side-peak suppression decreases with increasing feedback delay time. In addition, the linewidth reduction and side-peak suppression are relatively robust to the filter detuning, especially for higher feedback strengths and microwave frequencies. This is mainly attributed to the self-adaptive shifting of the red-shifted cavity resonance frequency to the center frequency of the filter in the FOF configuration.