Thevenin Equivalent of RC circuit
The Thevenin equivalent is found at a pair of nodes.  What it is connect to doesn't change. In this case, the black dots are probably causing confusion.  If you want the Thevenin equivalent made up of the source, R1, and R2 not including C then C doesn't change.  The pair of nodes are either end of R2, between R2 and C.  If you are only interested in DC then the capacitor doesn't matter much.

If you are interested in the frequency dependent Thevenin equivalent then C will make a difference.  You still have the option of not including it in the circuit and get a Thevenin equivalent of the source and resistors.  Then the the circuit will look like the one on the right.  If you include the C in the equivalent, you would end up with a source and a complex impedance without C across the output.