Currently, honeycomb cores are sometimes faced with an adhesive film between the Prepreg skin and the core. But you seem to be describing a viscoelastic midlayer.
This arrangement could be simulated parametrically with FEA and the polymer’s FEA Support Test Data. (Parametric FEA model: adjusting thickness and surrounding core/facesheet assembly material and geometry)
You could simulate the behavior you are interested in by performing a DMA temperature and frequency sweep on a viscous polymer you are interested in. Then, using time-temperature superposition principle, use WLF curve shifting to get the polymer’s dynamic properties in the acoustic frequency range.
Apply these properties to your FEA model and adjust (polymer thickness, core/facesheet geometry and material) as necessary to arrive at your desired design goal.
Drawback to this honeycomb modification:
One drawback to breaking the out-of-plane cell wall direction with a viscous polymer is the loss of in-plane Shear stiffness that honeycomb cores are particularly good at providing. The interface at the “hard” outer walls would need to move in unison or risk delaminating the midlayer and compromising your engineered benefit.
I was thinking about a way to not detract from the core’s structural purpose and achieve something similar:
What if “polymeric bubbles” were blown through hexagonal injectors into each cell. Blown in such a fashion to adhere to the cell walls and cure in place (likely with some inherent curvature).
Please contact me if you’d be interested in further research (firstname.lastname@example.org). I’m quite interested in this topic since reading your question.