There could be several things that need to be fixed to solve this issue. Go step by step to see if these steps fix your problem.
1. Login to WHM as root and check the hostname (Home »Networking Setup »Change Hostname). The hostname must be a fully qualified domain name (FQDN). In English, a hostname should consist of a name and a domain name. All valid hostnames should have 2 dots in them. Hostnames should never begin with “www”. Hostnames should also not begin with a digit, nor end with a hyphen (“-”). For example, if my domain name is example.com, the hostname can be xyz.example.com . If your hostname is not FQDN, some of the email servers will reject all emails from your server. You can change hostname via WHM or console. If you change the hostname, WHM will ask to create A record for it. You must create A record for your hostname. If this doesn't fix your problem, go to second step.
2. Some of the email servers (AOL, Live) check the valid rDNS data before accepting emails from another server. rDNS of IP of your domain must resolve to your domain. You can use rDNS lookup (https://remote.12dt.com ) to test that. If rDNS doesn't resolve to your domain, ask the hosting provide to fix it. Hopefully they will do it even if you have unmanaged plan.If this doesn't fix your issue, go to 3rd step.
3. In WHM, go to DNS editor (Home »DNS Functions »Edit DNS Zone) and check if you have MX record for your domain. If MX record for your domain is mail.example.com, there should be an A record for mail. If it's not there, add A record for mail. Also, make sure you have an A record for the hostname. If this doesn't fix your issue, go to 4th step.
4. I don't think this should be an issue as you are able to open your website, but there is no harm in checking. In WHM, go to resolver configuration (Home »Networking Setup »Resolver Configuration). If your hosting provider doesn't have resolver IP, use google's public DNS (22.214.171.124 - primary and 126.96.36.199 - seconday).
5. If none of these steps fix your issue, raise a ticket at cpanel.com.