Are you deleting messages by "move to deleted" (the trash can icon)?
And they come back after you check mail in the account?
Please check / fix the mapping of the app's "special" folders to server folders:
To restore the "mapping" of "special" (Sent, etc) folders to server folders, please do this:
- Long press the account
- Options and folders
- Folders
Let it refresh
- Find the Sent folder (under [Gmail]), tap its name, choose "Sync with [ Sent ]".
- Repeat for Deleted and Drafts, and possibly Spam and Archive (which Gmail calls All Mail).