Offers work differently compared to the rest of the service, and they are never mixed with your mail from governments or connected companies. Instead, these are gathered under the "Offers" tab and require your active consent. You cannot open or view circulars without giving consent to that specific store. Furthermore, you will not receive notifications for them unless you choose to turn them on.
Please note that the "New" indicator that appears once a week on the Offers tab cannot be disabled at this time.
If you no longer wish to see circulars under Offers, you can easily withdraw your consent while logged into the app under More > Settings > Senders > Offers.
If you have previously chosen to receive push notifications for specific stores, you can turn them off by opening an offer and disabling the notification toggle at the top of the screen.