Mail app on iPhone and iPad

A few months ago the mail app on my iPhone and on one of my iPads started behaving erratically. It will tell me that it is checking for mail, then that it is downloading a certain number of messages. It ten says it has downloaded them but nothing has been downloaded. Later it will download messages that are more recent but not any of the ones that it thinks it has already downloaded.

I am using a Gmail account for mail. The only way to get the missing messages is to delete the mail account and then reinstall it. In this way all the messages get downloaded. The crazy part is that this does not happen with my second iPad which continues to download all the messages every time.

It's very frustrating because I don't know when I am missing messages. As a double check I installed Outlook on the iPad that is erratic, set up my Gmail account and it is behaving perfectly, downloading all the messages. Has anyone had this problem before and can it be fixed?