Gmail accounts

have you made sure that the option in google security to allow non-secure apps is set ON? if it's off, only webmail and the gmail ios app can access gmail account.

From Google

" Change your settings to allow less secure apps into your account. We don't recommend this option because it can make it easier for someone to break into your account. If you want to allow access anyway, follow these steps:

1. Go to your Google Account.
2. On the left navigation panel, click Security.
3. On the bottom of the page, in the Less secure app access panel, click Turn on access.
   1.  If you don't see this setting, your administrator might have turned off less secure app account access."

If you have two factor authentication turned on, you also may not have this option.

https://support.google.com/accounts/answer/6010255?hl=en

You have tried the Gmail app? Have you tried deleting the Gmail app and downloading and setting it up again?