You are partially correct. You are correct about the part that the total voltage in the conductor is the addition of the voltage (S) and induced electromotive force, assuming that:
1. The applied magnetic field doesn't affect the flowing current due to the voltage (S).
2. The induced current due to the time varying magnetic field doesn't affect the magnetic field that generated it.
Taking those conditions into account will over complicate the problem, from your question we can assume they are valid assumptions.
The bit where you are wrong is when you say the voltage (S) is in series with the induced electromotive force. Please note that Kirchoff's circuit law (KVL in particular) is derived for cases where the magnetic field is static. For time varying magnetic fields (as the one you describe in this problem), KVL doesn't apply any more. Please have a look at the limitations section for KVL in this page. A nice explanation of why KVL breaks when there is a time varying magnetic field can be found in this presentation
EDIT: Lenz law (the law describing the generated EMF) is based on the magnetic flux variation, which could be either due to time varying field or due to changing area as well, or even both. So yes KVL would fail in the case of moving area as well.
In your edit, it is tricky to tell how the total current will behave. The top and bottom wires are connected to the power supply (S), so now you have three meshes in the circuit:
- A mesh consisting of V(S) which is connected to the wire to the LEFT hand side.
- A mesh consisting of V(S) which is connected to the wire to the RIGHT hand side.
- A mesh consisting of the RIGHT hand side wire and the LEFT hand side wire.
Now, depending on which meshes enclose the magnetic field, an induced current will flow in that mesh. For example, if the source V(S) is connected to the right of the shown figure, the magnetic field on the left wire will induce a current in mesh number 1 and mesh number 3, while the magnetic field on the right wire will induce a current in mesh number 1 and number 2. So in general, you need to do some calculation to find out whether the total current increase or not.
To make this concept clearer to you, please keep in mind that a current induced by EMF should always flow in circles in closed loops, which is a consequent of Lenz law, before you made your edit, there was no closed loops, so there was no closed loops and as a result there was EMF without a current. Now there are three closed loops, and you need to track the current in everyone of them to see their total effect.