This process can be done a few ways. it all depends on the type of access you have to the server. if you have just user access to the server and can not edit the DNS zones through WHM or SSH, the I would recommend to do the following steps:
1. Click on the Mail icon on the main menu of cPanel.
2. Click on the Modify Mail Exchanger (MX Entry) icon.
3. Click the Change an MX Entry link, and enter the domain of the new MX entry in the blank field.
4. Click on the Change button
You have now modified an MX Record through cPanel.