In order to make the best decision it is important to know how those two patterns are used elsewhere in Android applications. This will give us the best indication of what user's will expect.
Overall, aim to use the standards set forward by Google, while also remembering that they are guidelines, not rules. Make informed decisions about breaking from a standard.
Navigation Drawer
For reference the Material design guidelines for a Navigation Drawer tell us it intended for listing navigation destinations for your app.
"Simple Android menu"
I am not sure what you are referencing here, especially since we don't know what the "primary" nav items are that you haven't listed.
Menus
For reference the Material design guidelines for "Menus" tell us 1) it should not be used for primary navigation and if added to the Action Bar then it should be used as an 'action overflow menu'. This means you don't want to add top-level navigational items as much as it should extend on the action available in the current context.
Recommendation
If the primary nav items are permanently fixed in the app with Tabs or a Bottom Navigation Bar then putting other items in a navigational bar could work. Consider how other apps use it: They show all navigational items, including primary then you break the Android pattern.
Another path would be to use a Combined Pattern such as the "Side nav and tab combination" which would allow you to show primary items in tabs and then repeat those items and more in the Android Drawer.
The items you have listed should at least exist in a drawer and not anything smaller since they are important enough to reach.
