You mentioned powered speakers plugged into it, which means the output from the display is unamplified, and it is amplified by the speakers. So the wiring from the DAC, decoding the HDMI signal to the jack where the speaker is connected is analog and surrounded by very noisy digital circuits, especially power circuits, which induces the noise.

Most digital speaker systems, such as Bluetooth headphones have a single IC for both DAC and amp which effectively eliminates any induction of noise.

The weakest point in the audio system is the unamplified analog signal. Any induced noise is amplified with the intended signal.

The frequency heard is going to be a function of what signal is causing the induced noise. Depending on the display, it could be any number of subsystems. Usually related to the backlight, but not always. The noise is very likely completely unrelated to the audio being played.