"May we have more bread, please" would also be a good translation.
Personally, I'd never use "may" if I was asking someone to bring me something, only if asking for permission to get it myself.
Also "can we have more bread".
Why is "some more bread" wrong? (I wrote 'Could we get some more bread please').
Is this supposed to be asking, for example a waiter, for more bread or asking whether you're allowed to fetch more for yourself. From the English, I'd sooner assume the latter.
"Saisimmeko lisää leipää?"
"Saisimmekaa lisää leipää tuolla."