I would not recommend putting any money into repairing this pump. Here's why. When you look at the end of it, you can see one of the socket head screws is missing, and someone has filled that area with JBWeld. It sure looks like the housing is cracked near that JBWeld. This is called the 'pump head' and it costs about $150 to replace. You will not be able to get it apart easily and the replacement cost is more than a new pump! Add to that a check valve kit and seal kit and you can see it's not worth doing!
FYI, the "unloader" is the item on the top of the pump head with the spring. It is basically a relief valve that gives the high pressure water a place to go when you let go of the trigger on the wand. It's also used to set the max pressure of the pump. Down on the bottom, the item with the black plastic fitting is the thermal relief. If you run the pump without fresh water being run through it, it gets hot and this relief valve opens.
If the Honda engine is running OK it's worth putting a new pump on, IMHO!
FWIW....