The difference in the amount of processing required of subsequent occurrences following the efferent copies, as observed by fMRI-like technology, was striking. The idea is that this is one way the brain efficiently deals with speech recognition and variance. By (unconsciously) having "heard" the target or an idealized version of it just previously in the "mind's ear", so to speak, we have more processing power available to work on other things with . . .
Inner speech has been studied and employed in the second language research and practice extensively (e.g., Shigematsu, 2010, dissertation: Second language inner voice and identity) and in different disciplines. There is no published research on the direct application of efference in our field to date that I’m aware of.
The haptic application of that general idea is to “imagine” saying the word or phrase synchronized with a specifically designed pedagogical gesture before articulating it. In some cases, especially where the learner is highly visual, that seems to be helpful, but we have done no systematic work on it. The relationship with video modeling effectiveness may be very relevant as well. Here is a quick thought/talk problem for you to demonstrate how it works:
Imagine yourself speaking a pronunciation-problematic word in one of your other languages before trying to say it out loud. Do NOT subvocalize, move your mouth muscles. (Add a gesture for more punch!) How’d it work?
Imagine your pronunciation work getting better while you are at it!