Looks like a standard mATX Z370 motherboard to me. That small plate on the back should already be there, holding the CPU lock assembly in place. If it is not, then either you took it off or there is a serious mystery to be solved because it should certainly be there.
Installing that cooler should be a simple matter of removing the backplate that was used with the stock cooler, attaching the new backplate that came with the cooler, with the "arms" angled so they are down, closest to the board and the body of the backplate slightly elevated off the motherboard. Then attaching the cooler to the standoffs once they are fastened to the cooler.
There is nothing complex about this unless that board is extremely thick and I've never seen an HP board that was.
Keep in mind there are two ways to install that backplate. Flipped to one side for AMD and turned the other way for Intel.
There are different fasteners to be used depending on whether you are installing on an AMD or Intel motherboard, so make sure you are using the correct fasteners.