Seen these blotches before a few times. Last time i looked into one of these queries we narrowed it down to be a calcium carbonate deposit. Another previous experience was pyrite iron which had a more metallicy lustre to it.
Anyway. Both times, LTP Grout stain remover was the most effective. Its an acidic cleaner that breaks down both of these minerals. It will require a good scrub with a black emulsifying pad. This will take most off the surface but bare in mind there may still be some remaining within the slate that cannot be completey removed.
Go straight in with the neat solution. As soon as it comes into contact, it will fizz white. Scrub it, leave for 10 minutes, scrub again, rinse, leave to dry, repeat if required.
Once done it will need resealing.
Thats my method but
@Lithofin BOB is a genius at this stuff and may have some other solutions.