Klemen,
Sounds like a coupla possibilities. The first (and easiest) is ignition. It could be that the trigger coil/magnet or reluctor assembly may be weakened, providing little or no spark at kick start RPM's. Once the RPM is high enough (through pushing), the coil/magnet assembly may be just strong enough to then spark at those slightly higher crank RPM's. You can easily double check your spark trigger sensitivity by removing the plug, re-installing the spark cap on the loose plug, ground the plug body on the cylinder head or cases (to complete the spark circuit) and kick the engine over....slowly at first to see if you can trigger a spark at low crank RPM's. If there is no spark at low RPM, then kick harder until you do see spark. If you do find an RPM sensitivity to creating a spark, you may have found your problem.
The second possibility is low compression. This can easily make starting hard as well, but it's a lot more difficult to fix. It's caused by improper piston/cylinder wall clearance or sloppy compression ring fit or improper bore conditioning (poor ring seal) or poor valve sealing (intake or exhaust) or ALL of the above (or any combination). Get an accurate screw-in compression gauge, remove the spark plug, grab WOT with the spark plug cap GROUNDED and kick the shieeet outa it (about 5 to 10 kicks or until the needle no longer moves). Do this at least three times or until you see some consistancy in all of the gauge readings. I think you should see around 160 to 180 psi (for an 11:1 c/r). If you are only getting say 100 to 120, you gots problems....
Let us know what you find....and good luck!