Cleaning Terrazo floor tiles

Dear colleagues,
In our conservatory and downstairs lavatory we have a floor made of 9" square terrazzo tiles. This was laid before 1976, and on the whole it is fine. I clean it with Ecover floor soap, and this has improved it a lot.
However, there are several places where the surface is marred by smears of cement from when it was installed. I have tried getting these off with an ordinary scraper, of the wallpaper sort, but no progress. I know that removing the cement will do nothing to improve some marks possibly caused by rotary tools or cold chisels. Short of replacing the whole floor how can I improve it?
I have 1 spare tile, without any maker's name. It seems to be in 2 layers, the top one being polished and the underneath one is more like asbestos.
TIA, Robin.
Robin Phillips,
Beeston, Notts., NG9 1BT
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