This is a difficult one! Scuffed bumper plastic is very difficult to put right.
I'm still a little unclear as to the paint situation. Is the bumper painted or just black plastic?
If it is painted (using the same paint as the bodywork of your car) then I would recommned a fine polish like Autoglym Super Resin Polish. It sounds like the scratches are very fine and using T-Cut might do more damage. You will have to put a fair amount of effort when using the Autoglym product but the results will be worth it. I would then recommend waxing the polished bumper or protecting it using some other high gloss finish.
If the bumper plastic is not painted then getting the scuff mark out is going to be more tricky. You can of course cover it up temporarily using products like 'Back to Black' but these are just temporary measures as the colour soon wears off and the white scuff mark will reveal itself over time.
Also, if the bumper is black plastic (not painted) then is it smooth black plastic or textured? If it's smooth then it's probably a little easier to treat than if it's textured plastic.
I'll get back to you depending on your response.
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