You can make a simple check on the fuel getting to the carburetor by disconnecting the fuel line at the carburetor end, positioning it close to the normal carburetor connection, putting a cup under the end of the hose, and turning on the fuel tap. If it runs a decent stream, you can eliminate the upstream bits like the line, tank, tap and in-tank strainer (if it has one).
If you have fuel to the carburetor, you could check a couple of basics like compression, spark, air cleaner and governor. You already said that visually the governor looks damaged, and this needs to be sorted out immediately. I don't like the sound of it taking several pulls to start: it shouldn't. And why did the pull cord break?
