Hello poppydoc,
To help resolve this issue where your Gmail isn't syncing between your iPhone and MacBook Air
You’ll also want to quit the Mail app on your iPhone and iPad, and then restart the devices. Follow the steps in these articles to do so:
Restart your iPhone, iPad, or iPod touch
How to force an app to close on your iPhone, iPad, or iPod touch
How to force an app to quit on your Mac - Apple Support
Shut down or restart your Mac - Apple Support
It might help to remove the account from both devices then add it back again. You can find more information about adding an email account to your devices below:
If you continue to experience issues after removing the account and adding it back or if the issue is only with your Gmail account, we'd recommend contacting Google directly for assistance.
Best Regards.