To remove a contact you must first go to the Contacts section within the client record.
When you click on the contacts section, it displays the contacts below. See this article for more information.
To remove a contact from a client there is a Remove button at the very end of their record as you can see in the image below.
When you click Remove, a validation message appears. If you click Yes the contact is removed and if you click No the contact is not removed.