There are two kinds of exotic cable manufacturers. The normal ones and the supernatural ones. Most of them don't claim that their cables will clean the power supply, because that would be ridiculous and seen by anyone applying logic of KMs of dirty cables and the last 1 meter doing magic.
The normal ones are the PS audio kinds who claim their cables have excellent shield and the power cables do not emit EMI and hence do not cause degradation of audio signals in other equipments feeding audio to the main amplifier.
The legendary ones claim their cables are immune to triboelectricity. Tribolectric charges become relevant in the medical field where impendance of some equipment is extremely high (much larger than what is there in the audio industry). Even a small amount of charge from the cable polymers rubbing or moving will cause readings to change. IIRC,
furutech is the supernatural legend here.
There is a video from ASR
here and a rebuttal from Professor who has been rebutted in the ASR video
here. You can also read
this on Archimago's Musings on the above Professor hard to believe papers on cables and I would end this with Archimago's own words