Gmail and Mail on iMac

My husband's iMac seems to have become selective with Gmail! He has a Gmail email address and uses Mail on the iMac as his email programme. However, recently he has told me that he hasn't received a message we know has been sent to him. This happened this morning with an important message which had, fortunately, sent to both of us. I've checked all his account settings for Mail and can see nothing amiss. I then logged into his Gmail account via Google Chrome, which he uses as his default browser, and there was the missing message. Any idea why this might happen, and how I can correct it would be most welcome! or is my only option to try and help him to use the Chrome to Gmail route for all his emails? Very many thanks in advance!