When connected to bluetooth headphones there are several issues that almost always occur when I ask the Google Assistant to "read my messages":
1. If the phone screen is OFF and If I have no messages it says "something went wrong" - but not through the headphones.
2. If the phone screen is OFF and If I DO have a message it says "something went wrong" - not through the headphones. But then it says "do you want to reply" - through the headphones. Note that it never actually read the message.
3. If the phone screen is ON when I ask to "read my messages" it will sometimes work correctly. Other times it says "Something went wrong".
I have posted to Google's forums and the feedback I get is that this is an issue with the phone.