Can you/should you mulch the pockets of a GreenStalk planter? Yes, and yes.
Avoiding bare soil is just as beneficial when growing in a GreenStalk as anywhere else. Mulching helps the soil retain moisture and avoids the hard crust that can form of hot dry days. It helps minimize weeds and gives some protection for temperature extremes.
There are many kinds of mulch - grass clippings, shredded leaves, wood wool, paper, sawdust and compost to name a few. Note that woody materials like wood chips and bark nuggets should not be used in a vegetable garden, as they rob nitrogen from the soil.
Mulching is most often done when the weather becomes warmer, as it tends to cool the soil and that is the last thing we want in early spring!
It is advisable to allow any direct sown seeds to germinate and grow a bit before adding mulch. Shown below, I had recently sown carrots and beans in one of these pockets and waited a few weeks to make sure the carrots had sprouted, and do any necessary thinning, then it was time to mulch. We have the tiniest of lawns which would take a whole summer to provide me with enough grass clippings for a tower, so I like to use wood wool.