The little black hose is a vent. The pump diaphragm needs to be vented to the atmosphere for the pump to work properly. Used to, before EFI was popular, I was playing with turbocharging a Ford engine that had a carburetor and a mechanical pump, which is the same basic principle as the Kawasaki pump. The biggest problem was fuel pressure and volume. Volume was solved by using a bigger pump. Pressure was still an issue. So I devised a solution. The pump had a vent hole on the body, so I drilled the hole and tapped it with a 1/8" NPT thread, put a 1/8 NPT-1/8 barb fitting in it and ran the barb fitting to the turbo's discharge side, but before the throttle body. Worked marvelously, WAY better (and more reliable) then a boost referenced fuel pressure regulator and electric pump.
Anyway, your Kawi's vent line should not have any fuel in it. If it does, the diaphragm is ruptured. Also, make SURE that the vent hose is routed away from the muffler. It doesn't go to anything, just the atmosphere. In factory configuration it was dumped right on top of the muffler area; and if (when) the pump diaphragm ruptured, gasoline would be pumped directly onto the top of the hot muffler causing a fire. All you do it route it down the side of the frame somewhere and let the end of it dangle in the wind somewhere away from the muffler.
A couple thoughts on the bogging issue under load. One, I'd suggest replacing the factory fuel lines. The come apart internally and block everything. It's easy, pull the tank and replace the lines. While the tank is out, pull the pickup tube out and blow it out. If anything comes out of the tube, you'll want to look into what's in the tank to plug the tube, and possibly clean the tank out. They were pretty commonly plugged at the elbow where the pickup tube meets the fuel line. While you have the tube out, cut a vee notch into the very tip of it. That or cut the end of it at an angle. This helps keep from plugging the tip of the tube if anything gets into the tank. And thirdly, I have run into a few 1860's and 1760's that would acually boil fuel inside the fuel line. The lines were routed too closely to the engine and muffler causing fuel to boil after running a while. Just re-route the lines a little further from the heat and it will be fine. This is more pronounced in the late spring/early summer as the change from winter blend gas to summer blend gas is taking place. Many folks will go out & buy 5-10 gallons of gas and it'll sit in a can for a while until it's time to re-fill the mower(s). If it's winter blended fuel, it can boil at as low as 80°F.
