According to another post by jhwir , puis is an archaic form of the first person present indicative peux. It is still used in inversion or with the conjunction si .
Puis-je aider? (Can I help you)
Si je puis permettre (If you don't mind)
Yes very old fashioned and no body says anymore. They should change it!
"Could I speak to you?" or "Could I speak with you?" Both should be accepted.
it is not silent but it is not separated from "puis".
So you should hear "Puij" and not "Pui je". (and if you pay attention you hear it there)