Remove the client name field from Programs. This is an unneeded field and can create issues should you ever need to edit a given client name.
In the Client to Programs relationship, enable "Allow creation of records via this relationship" for clients.
Put the Name field from clients on your programs layout. If you create a new record on the programs layout and enter a client name, FileMaker will a) create a new record in Clients, b) generate a value in ClientPrimaryKey (assuming that it's an auto-entered value such as a serial number) and c) copy that value to the ClientForeignKey field in programs.
Thanks very much, that made it all clear