cPanel/WHM Website Mail Issues

We have a Linux dedicated server which we provide hosting to a number of clients on. Some clients have said they can send and receive mail fine to domains hosted elsewhere, but that they have trouble sending mail between domains hosted on that server, as some client have multiple domains hosted on our server. We also have some client whose mail isn't hosted with us (we only host their website) and they say that mail sent via their websites aren't being received. I believe the server thinks we host the mail or doesn't know where to send the mail or something. I need someone with experience to fix these issues URGENTLY and we are interested in maintaining a contract with this person to provide continual maintenance/support for our server ongoing.