Lenovo may also have a "battery optimizer" built-in, same as Huawei does for example, or MIUI.
It's best to check with their support -- is there something in this phone which tries to optimize the battery by restricting apps in the background?
I don't happen to own any Lenovo's sorry.
Re: Doze mode
This is also a "battery optimizer" but built into Android itself.
AquaMail should have prompted you to "whitelist" it there, but please check app settings -> mail, receiving.
If not already done, there will be a panel / button near the top there, prompting you to "tap to exclude from Doze Mode" (please do that).