Comment notification with split web/email hosting

I have a personal WP blog via HostGator’s shared hosting; v4.9.1 and the Suits theme. The site is configured to notify me by email when a comment is posted but so far I have never received such a message. This is not a huge deal because there are very few comments but it would be nice to get an email when the occasional one does come in.
Additional observation: If I try to change my administrative email address, I don’t get a confirmation email there and the change doesn’t go through.

I am wondering if this could be due to my using a separate hosted email service instead of HostGator as my email provider; e.g. no outgoing mailserver is configured so WP can’t send mail. If that is the case, is there a way to tell WP which SMTP server to use? I don’t see a way to set that up the SMTP address within the cPanel site administration tool.

Thank you.

The page I need help with: [log in to see the link]