Gmail won't retrieve mail on iphone 4 in Mail App

For the past month, one of my 3 gmail accounts has stopped retrieving mail in the MAIL app on my iphone 4. I did a workaround by downloading the Gmail app for the one account but now the Gmail app is signing me out several times a day and it's well, annoying.

I've updated the OS, reinstalled the gmail account in question but it won't update in the Mail app. For info, I have two-step verification on the account but also on another that works fine.

Any ideas what could be causing this? My two other Gmail accounts in the Mail app update and retrieve mail fine.

Thanks !