I assume you are refering to a moving circuit in a fixed magnetic field when you are talking about a 'motional emf'. This emf is induced by the change of magnetic flux through a loop.
The induced emf will oppose the change of the magnetic flux that generated it, i.e. it will induce a magnetic field which is opposed to the external field responsible for the change in magnetic flux.
This seems to be the point where you get confused. The induced emf generates a field and not a constantly changing magnetic flux. But as you stated in the formula, only a change in magnetic flux will result in an emf. A constant magnetic field does not induce an additional emf. Your title question is "Does the magnetic field due to the current from a motional emf induce a counter emf?" The answer is "Only a change in magnetic flux can induce an emf."
So for the bookkeeping of the net emf, it is only required to track the net change in the magnetic flux. In most simple examples, you will either have a constant magnetic field or a constant area of the loop. If you have a fixed magnetic field, keep track of the change of the area, and vice versa.
More on the moving circuit and the resulting emfs can be found here.
You might also want to read up on 'inductance' and Lenz's law.