How do you say go to your mother's house in jamaican?

One possibility: "Können wir zu dir nach Hause gehen?"

I know what you mean but she's only doing that to protect you.

The Spanish phrase for 'I am going to the supermarket with my mother' is 'Voy al supermercado con mi madre'.

¿Puedo ir a su casa manaña?

If you mean "friends' house," you say, "¿Puedo ir a la casa de mis amigos?" If you mean "friend's house," you say, "Puedo ir a las casas de mi amigo/a [depending on your friend's gender]?