You can cut the dead stuff off, but be sure its dead and make you cuts just above the node. You have your plant planted too deep in the ground, and with the extra mulch, its holding on to too much water. That's the problem with your Hummingbird Summersweet. Remember to only use mulch in the dry season when you get no rain and the temps are hot. If you use mulch in the rainy season the rain water and will not have a chance to evaporate and suffocate the plant from lack of oxygen.
A plant pushes water through to the edge of its leaves, and if given in excess, this can cause veins at the edges to burst, which leads to browning.