1. First and Foremost is fuel. Fuel should be less than 2 weeks old (unless it's been properly treated, and how to do that is not the focus of this reply). Fuel that was left in the unit over winter cannot be trusted. As a best practice: Never, ever store any small engine with fuel in the tank or carb.
2. If you can get the unit running, as per your description, then often it is the needle jets that gum up. With the unit running, turn the adjuster screws back and forth and often times the moving fuel will wash the jet clean. Be sure to re-set the screw exactly where it was. There are most probably 2 screws marked Hi and Lo, which refers to the engine RPM they control. The Lo screw usually has the biggest effect on cranking.
3. I would suggest you start using a dual action stabilizer and cleaner additive in all your small engine fuel. You can treat the big can you fill all the small engines from and they will all benefit. Such an additive would be StarTron. The action described in #2 will be faster and more effective if you do.
Hope this helps.