Is this the type of mower that you engage the blade when needed, at the handlebars? If so, they do pulsate a little when the blade is off. Try it with the blade engaged.

I don't think that's the problem. I can only start the engine with starter fluid. Any kind of blade clutch engagement instantly kills the engine. Thanks for the idea.
But, it seems like the self-propelled and blade clutch are working.

I had not seen varnished Honda carbs until recently. They seem to have escaped this dilemma. Here is what you're looking for.

Most likely, you'll have to remove the carb. Make careful note of the linkage and how it comes off. Remove the bowl nut and in the center of the carb you'll see an opening just under the bowl nut. You'll need a special, thin screwdriver to remove the first jet. It has a slotted head with a center jet. Above that is the long tube with many small holes. Both pieces have to be thoroughly clean. Use carb cleaner to clean the rest of the carb; any orifices. Do not overtighten the jet when you put it back in. Very light torque; it isn't going anywhere. This should solve your problem. This carb worked like a champ when I was done. Good luck.

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Thanks for the info Baymee. I soaked my old carb overnight and did get a lot of junk. The problem is I purchased a new carb from Honda. I didn't tincker with anything on the new carb. The new carb did exactly the same as the old carb.

Have you drained and cleaned the fuel tank lately? Dirt might be getting into the carburetor from the fuel tank--this (obviously) would affect both the old carb and the new one.

Tarheelman, I was just thinking about that. I will do that tomorrow. thanks all for the input

If it was a restricted fuel supply, the mower would probably start up and run fine for the first minute or so and then start pulsing. Is this the case?

I flushed the tank, hooked the fuel hoses up again. I turned on the fuel shutoff valve and left the fuel line off the carb. Same result. It seems like there is fuel coming from the gas tank to the carb. Maybe not enough fuel is coming through. I don't think the line is pinched. I even put some carb cleaner in the fuel line. After the engine starts using starting fluid, it will kind of run for quite a while with the choke set fully on. As soon as I try to take some choke off is when it starts to pullsate and eventually dies. Strange stuff. I'm not sure about the engine yet - valves. I'm using the new carb I got from Honda.

baymee,
Yes, that is the case. Maybe not enough fuel coming through the lines - or maybe a gasket leaking problem.

If the fuel tank and lines weren't cleaned as the new carb was installed, it is time to clean the new carb. Straight cleaner in the lines is not a great idea and can deteriorate the line. Also, if there are deposits of old fuel on the valves, they will stick as they try and close. Normally spraying cleaner in through the venturi as it starts to up and run will lessen them enough to run pretty well.