Can't update my mail password on Macbook Air
It looks like you are trying to send out emails from your Gmail account using Apple's iCloud as SMTP server which is not correct. Here's what you can do: 

Open Mail > Preferences > your Google account > in the dropdown menu select Outgoing Mail Server (SMTP) > Edit SMTP Server List

Ensure that the settings match the following:


Outgoing Mail (SMTP) Server - Requires SSL/TLS: smtp.gmail.com
Port: 465 or 587
Requires SSL: Yes
Requires authentication: Yes
Full Name or Display Name: [your name]
Account Name or User Name: your full Gmail address (username@gmail.com). 
Password: your Gmail password


More info on Google Support Docs: Change SMTP settings to check Gmail on other clients