Constrain certain variables in CNF to unique satisfying assignments
If you could add clauses to a CNF formula to cause it to be unsatisfiable iff certain sets of variables don't have non-unique assignments among the formula's satisfying assignments, you could use this same method to solve UNIQUE-SAT for any CNF formula.  Since UNIQUE-SAT is co-NP-complete, you'll have proven NP = co-NP.  So it is unlikely there is a solution to the problem that involves simply adding clauses.