"Table or view does not exist" on synonym
Synonyms have nothing to do with privileges.  They are simply a way to simplify naming.

The error you are getting appears to indicate that user_app does not have privileges on the user.clients table.  You'd need to grant that

GRANT SELECT, INSERT, UPDATE, DELETE
   ON user.clients
   TO user_app;


Of course, in your actual system, I'm guessing that there is a role that you would grant privileges to and that role would be granted to user_app.