It's a bit difficult to be 100% confident with an ID given the leafless state of some of your plants, but I feel pretty good about saying Hygrophila corymbosa 'angustifolia'.
As far as pruning goes, I usually avoid trimming leaves, especially if the plant is new or adjusting. This is because even unsightly leaves can do photosynthesis, which the plant needs to make new leaves and/or recover from the problem that made the old leaves raggy. If your plants are growing well and you bottom out on phosphate and got holes, then I wouldn't worry too much about trimming a few. With stem plants, it's generally a good idea to cut their stem, making sure each half has at least a few nodes with leaves (for the aforementioned reason), and replant the top. The bottom should form two new tops.