That is a 4 cycle engine. If you can get a carb kit might try that. I have a 20 hp Kawasaki engine on a mower as I put a carb kit in and it did not help much. A new carburetor was $200. I found a new carburetor old stock carb on Ebay for $100 and I have probably put 300 hours on since, running like it should.
Here is a carb for yours the same price as your quote. They have sold a bunch. You might take a can of carb cleaner and clean the one you have and help as I did not see a kit for it. Running lean it is getting more air than fuel.
https://www.ebay.com/itm/Honda-16100-ZE3-V01-Carburetor-fits-GX340-GX-340-Generator-Engine-Motor-OEM/351719109483?epid=1100281619&hash=item51e417b36b:g:rvUAAMXQrvpREtue