G’day Ryan
Does water flow through the pressure hose when the machine is connected to the mains?
That pump is “high speed” pump - it turns at the same speed as the engine (the other type is “low speed” which use some type of reduction system (gearbox or belt with different size pulleys)). This type of pump should be pressure fed from the mains (if there is enough volume), or a feeder pump, or a tank mounted high above the pump.
Regarding the leak - there shouldn’t be any leaks from the hoses, pump or unloader.
The unloader has springs and o-rings inside and the o-rings do wear but this doesn’t doesn’t usually leak to the outside, is it leaking from the top?. The unloader can, and should, be serviced - replace o-rings as needed and lubricate with a water based silicone tap grease. There are also rebuild kits available if needed, but unloaders are not expensive to replace.
The black hose is the “bypass” hose. The pump pumps water continuously and it must go somewhere, either out the high pressure hose/gun or back into the inlet. When the gun is released the water stops flowing, but the pump is continuing to pump water and it must go somewhere, so the unloader reacts to the increase in pressure and opens a port which allows the water to keep flowing (in your machine it goes back into the pump).
What is the white hose that loops behind the machine? Is it a chemical injector?
Regarding servicing -
The pump and engine use the same type of oil (usually a 30 wt non-detergent. Do not use regular car/truck engine oil). Change them regularly.
The 2 sets of 3 nuts on the brass pump head can be removed to reveal the valves which can be removed - these have a plastic frame so be careful - to check for corrosion and debris which reduce the performance of the pump (replacements are available).
The brass head/manifold can be removed to reveal the plungers (there is no gasket). These should be cleaned and lubricated (water based silicone tap grease).
Do you have a water filter on the inlet? get one with a clear bowl to see any debris that come from the hose, and keep it clean.
Fuel system -
Unscrew the float bowl (bottom of the carburettor) and remove for cleaning, plus there may be a small bowl nearby that also catches debris in the fuel line - remove it and clean. If there is a lot of fine dark sediment in these you may have rust in the bottom of your fuel tank - if so, clean these regularly and consider cleaning the tank and even using a liner/paint to fix it.
If you need to pay a shop to do all this - ask if you can watch as you need to be able to do all this yourself in the field and at home.
Hope this helps
John