Actually, I would first look through your contract with your provider and see if there is anything in there that says they will be reading your email. And if there is, find another manager. IF there is not, then you ask why your manager is violating your privacy? Then threaten a lawsuit. Then get another manager.
I manage content on a couple of small websites, and there is no reason I would need to read their emails. None. Zero. If there is a problem with the website, they let me know. If there is something that needs to be changed, they let me know. The only reason that I would need to read emails going to my customers was if maybe I needed to help fill orders, or answer questions about the business, but then I would no longer be a content manager; I would be a business partner.
So here is the deal. You are a customer. They are providing you a service. You need to ask them why they are getting a copy of your emails. If you wanted to be less confrontational, you would say "I notice from the logs that it appears that a copy of all my emails are going to account x. Is that true? Who has access to that account? Did I authorize anyone to do that? Do you have a good reason why you need to do that.? If any of the answers are unsatisfactory, find another manager, change all the passwords to the site, and then fire the current manager. Do not let them know they are going to be fired until they are locked out of your website.
You could change the configuration on your own, but without knowing what your setup is, I couldn't help you with that.
not wishing to hurt his/her feelings
Screw their feelings. You should be indignant.