I am guessing that Contacts section is a related list section on the standard SFDC Contacts object. What is the Client Contacts section? Is that a related list section or another section? If it is a related list section, then what object is it based on?
We show the Add button on all related section on SFDC objects and the Add button should show up if the user viewing it has permissions to create records on that SFDC object.
We have plans to bring in similar capabilities to related list sections build on MDA objects as well.