iOS 9.3 is coming in the Spring and will introduce Night Shift as well as a number of other new features.
Apple Acknowledges Continuing Issues with OS X Mail, Offers Manual Workaround
Some users found that quitting and restarting Mail forced the application to download new messages, but Apple has now published an awkward workaround that allows the app to stay open while still forcing a download of new messages.
OS X: New email messages not received until Mail is quit and reopenedThere is no indication of when the bugs with Mail.app will be fixed, though beta releases of OS X 10.9.2 ask developers to test Mail.app for issues.
Quitting and reopening Mail forces it to re-establish a connection to the email server. You can use these steps to receive new mail messages without quitting Mail:
1. Choose Mailbox > Take All Accounts Offline.
2. Choose Mailbox > Get All New Mail.
As a shortcut, you can also add the Take All Accounts Offline and Get Mail buttons to your Mail toolbar.
1. Choose View > Customize Toolbar.
2. Drag the Take All Accounts Offline button to the Toolbar if it is not there already.
3. Drag the Get Mail button to the Toolbar if it is not there already.
4. Click Done
To receive new mail messages, click the Take Offline button, then click the Check Mail button.