The denial of, mind to brain influence, is born from Epiphenomenalism's materialistic roots. Epiphenomenalists do not in principle (want to) accept the mind as a distinct substance, that is, they would reject dualism were it not for the difficulties in explaining mental properties.
With this in mind let us explain the mind, from the Epiphenomenalist perspective, using an analogy. We equate the mind with a computer screen, the brain is then the CPU. Note, we are thinking in terms of the temporal/events shown on the hardware of the screen, and likewise the software environment of the CPU.
Everything on the screen is caused by the CPU, but not everything the CPU does goes to the screen. The screen can think that it is doing/influencing what happens on the screen, because there is a procedural logic to events on the screen, that also happens to match the underlying logic that constitutes the screen content. However all computations are actually done by the CPU.
Now the mind may think it is deliberating, while in fact it is the brain that does the data processing, writing the process and results to the mind almost in real time. The mind thus mirrors the brain's activities, some of which are giving structure to the mind and filling it with content. Therefore the mind, its properties, its content and events, are all epiphenomena, everything the mind 'experiences' are mirror/phantom processes. So what happens in the mind, stays in the mind.
Of course the criticism remains: What is the the mind for?