Download Minecraft PE 1.16.40 FULL VERSION for Android you can in this article. Unfortunately nothing were new. Mojang only fixed 4 bugs. Nevertheless I highly recommend you to update this version. It also fixed some crashes on iOS.
What will be new in v.1.16.40?
As I said Mojang fixed only 4 bugs. List of this bugs are below:
Fixes:
Note that you can also share with Mojang your bugs or crashes. Just open official and write them.

What will be new in v.1.16.40?
As I said Mojang fixed only 4 bugs. List of this bugs are below:
Fixes:
- Fixed bugs on Nintendo Switch
- Fixed bugs on Playstation 4
- Fixed an issue with account sign in button on Android
- Fixed some crashes on iOS
Note that you can also share with Mojang your bugs or crashes. Just open official and write them.