There have been several patents where magnet assist has been a feature. So yes and no, because I don't know. The video could be a fake or it could be a phenomenon I am not aware of.
Your knowledge and comments would be most welcome.
Unfortunately, in this application the use of the permanent magnet on the transformer core does not improve
overall efficiency. It brings back some of the core loss due to saturation caused by the DC bias the applied square wave inherently creates. How this happens:
The comment under the video says the transformer was fed with a single polarity 50% duty DC square wave.
Such square wave has a resulting DC current component which biases the transformer core i.e. it shifts the operational point on the B-H curve either above the center (zero) or below the center point (depending on the direction of the current).
In the 1st attachment you can see the positive half of a typical B-H curve for most ferromagnetic cores and points P, Q, R and S represent different operation points i.e. core excitation levels with relative numbers on the horizontal axis and the numbers accidentally nearly correspond to the values shown by the Ampermeter.
In the no magnet attached case, the operational point should have been between point R and S, this is the start of core saturation area and the current was little higher than 4 Amper on the meter.
With the attached magnets, S - N polarity first, the current increased to 6 Amper: this means the magnets added their flux indeed to the core but with a polarity which shifted the operational point close to point S where saturation was almost full, i.e. core permeability decreased further on.
When the magnets were flipped and attached with N - S polarity, the current decreased significantly to little higher than 1 Amper, the operational point should have shifted towards point P, a good move into the linear part of the B-H curve. This magnet polarity restored the effect of the presumably unwanted DC bias the single polarity square wave willy-nilly created, and initially caused (together with the motor as the load) the shift towards point R and beyond, i.e. towards core saturation.
So the use of permanent magnets in this particular setup showed improvement in efficiency with respect to the no magnet attached case
but only in the sense they compensated the unwanted saturating effect of the single polarity drive current.
AND consider this:
If a bipolar square wave is used instead of the single polarity (unipolar) square wave, then there would be no any DC bias imposed on the transformer core. Such a bipolar drive signal can come from a normal H-bridge for instance, the signal changes from a positive peak (+Vp) to a negative peak (-Vp) while crossing zero value too. Just like a normal sine wave with zero crossings between +Vp and -Vp amplitudes.
The amplitude of the bipolar square wave excitation current a H-bridge can provide would optimally be chosen to reside around operational point Q which is the center of the linear part of the B-H curve (the permeability of the core is the highest at point Q, anywhere else on the curve it is smaller than at point Q).
Of course, the B-H curve continues into the negative values of the co-ordinate system, see the negative curve part in Figure 1b in the 3rd attachment, (so there are similarly labeled points on the negative side of the curve) and note that the simple B-H curve does not show any hysteresis loop for simplicity.
I attach a drawing on single polarity (unipolar) and bipolar square waves to help understanding, and this link
https://electronics.stackexchange.com/questions/299571/how-to-calculate-the-average-power-for-a-square-wave-dc-signal deals with power dissipation calculation in a normal 1 kOhm resistor driven with unipolar or bipolar square wave (duty cycle 50 % for both cases).
I attach a drawing taken from this link
https://www.mdpi.com/1996-1073/15/23/8842 , it nicely shows the effect of DC bias (for a sine wave excitation example) on transformer core saturation, causing the current to increase beyond its normal loaded case value.
Yes, there are several patents or patent applications where magnets are said to assist and / or attain certain advantage in operation, for instance their 'free' flux is added to the flux of an electromagnet (Hildebrand, Flynn etc.) Such flux addition (if that is the case) surely works and may increase overall performance but the main question is how to tackle the problem the increased Lenz effect can cause? i.e. increased flux surely increases induction or torque etc but Lenz law does not care how the higher flux was created, it reacts on it with higher counter emf or force etc (action - reaction).
So the video is not a fake, it shows an inherent core saturation caused by the single polarity excitation of a ferromagnetic core that had a closed magnetic circuit.
Gyula