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'Play' in a brake pedal is difficult to define and I can't find the link to your video.
If it clearance between the operating rod and the master cylinder then you have an issue with adjustment.
If the movement is due to expansion of the rubber brake lines then you may be able to improve this aspect of the braking by either fitting new lines or PTFE lines.
Have you recently change pads to a new type?
It could also be due to slightly warped discs causing pad knock off.
is this a recent change or have you had this problem for some time?
If, however, you make the pedal too solid you could end up with very 'snatchy' brakes that will lock without much warning.
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