To remove a child:
On the persons profile page, click Edit at the top.
On the right hand side, where it says Children, click the add/edit.
Don't enter any details, just scroll down to the very bottom to Remove Child:
Click the button next to the child you want to remove.
To remove a spouse it's almost the same. Next to Spouse on the right, click the add/edit button. Scroll to the bottom until you get to Remove Spouse, check the one you want to remove.