Why does the Honda GX160 engine start to pulse after running pressure washer for a few minutes then finally kills?

If you just want to check the engine then starting without water is ok but don't run for very long as the pump seal will get hot and may end up leaking when you finally do start to pump water.
Here is the link to the manualy;
http://redlionproducts.com/media/41984/honda-gx160-owners-manual-37z4f603-r1.pdf

Rebuild the carburetor. Just replace the old gaskets, diaphragms, needle and seat. You can get a rebuild kit from your local lawm mower repair shop. also, don't run last years gas and if you keep gas in the tank over the winter buy a winterizer and pour it in the tank when starts to get cold. Hope this helps!

Being petrol driven, it may have a mercury kill switch which puts a ground on the ignition circuit stopping the engine if you move it. Perhaps for safety in case you overturn the unit somehow.

Ii know you must have replaced the fuel filter. Did you replace the fuel line? It is obvious it is not betting enough fuel or it is out of time. Have you tried adjusting the jets on the carb? Have you checked the timing? There is a crankshaft key-way that may be bent just enough to get the engine out of time. And finally have you checked the fuel pump which I would suspect is your problem. The diaphragm may be busted.

what about the linkage between the governer and the carb. Is that working and not binding?

The reason it stops spraying might be a clogged gun, hose or nozzle-tip.
The reason for pressure buildup killing the engine might be a stuck unloader valve in the pump.

While you are checking each component of the "output" side, spray some penetrating lube into the "input" side of the pump (where the garden hose connects) and let it be working on the unloader valve.

Fix the spraying problem first:
Start at the nozzle and verify that it is open by using low-pressure air. Clean or replace as needed. Next check the hose (with nozzle and gun unhooked) and finally the gun/trigger mechanism.
All must be clear and open to perform correctly.

Next, pour a few ounces of power steering fluid (or similarly LIGHT oil) into the garden hose, attach it to the pump and turn on the water supply.

Leave the "output" hose, gun, etc. UNHOOKED and point the output port of the pump (where the high-pressure hose connects) towards a vacant, safe area.
Startup the engine and allow the oil in the garden hose to work its way thru. (About 1 -3 minutes of run-time.) Shut it down and attach the high-pressure hose.
Hope this solves the problem!

It could be several things...oil level, make sure its sitting on level ground...if not did you use StaBil when you laid her up for the season? If not, go to Advance Auto and buy "SeaFoam" and following directions run it for as long as it will run...make sure you have water running through the pump during tests!
Good Luck

Most Honda engines have an oil pressure switch that prevents engine damage due to low/no oil pressure. Make sure there is sufficient oil in the engine and that the wire is connected to the pressure switch.

Buy a new plug, drain out all of the old gas and put in new gas, and take a look at the air cleaner. If the air cleaner is dirty replace it. There you now have a good spark, good fuel and good air. The engine should start. If it doesn't get some carberator cleaner spray and some starting fluid spray.

Pressure washers work under very dirty conditions, no wonder they encounter engine problems. Here are a few pointers from a former pressure pump tech. The unloader is a source of engine overloading however this can easily be checked with a pressure guage attached to the gun with a tee fitting having the guage and the normal tip used for the unit. The pressure should never rise over rated pressure. The pressure should drop considrably when trigger is released. The life of the pump and engine can be increased by setting max pressure to 75% of rating. Most engine problems are from following conditions:
Leaking or misadjusted valves; clogged muffer /spark arrester; weak spark coil that is about to fail completely; kill switch/oil sensor wiring and or sensor mal function; gas cap not venting properly causing vacuum in tank and wet or dirty air filter.
All these problems easy to check, fastest repair is to disconnect oil sensor wire to check oil safety system. Then temp replace coil with known good coil. Loosen muffer bolts to check for clogged muffler. Temp run engine without gas cap to check for vacuum. Finally if everything fails, run the engine with carb cleaner spraying directly into the intake. If the engine continues to run with carb cleaner then the problem is fuel delivey and faulty carb. enjoy