You can turn this off in AquaMail:
Long press the account -> options and folders -> account options -> disable "remove deleted messages".
You will also need to make sure that AquaMail fetches messages completely when checking mail, so it doesn't try to go back to messages that have been deleted on the server.
You'll need to do this in two places:
- App settings -> Network (under both WiFi and mobile) -> load when checking mail. This is for message text.
- Long press the account -> options and folders -> preload. This is for embedded images and attachments.
The above should do what you want, although it's not the best scheme (talking about the entire setup, including the too-frugal server and the too-eager desktop mail app).