I've gone through the same problem on both my GX-35 and GX-50.
I haven't figured out a way to make it work yet. Basically, you've got 5 choices for Honda GX's.
1. You can either order a new custom made cable from Staton. Easy
2. Find a new adjustable barrel throttle screw with a larger inner diameter. Not so easy
3. SLIGHTLY dremel the lead throttle cable end, so it fits through your existing adjustable barrel throttle screw. Sorta easy...if you don't screw it up
4. Fabricate your own. Not so easy(the 1st time)
5. Go back to using the original throttle. Easy
....I ended up choosing #4, and can now make my own throttle cables whenever I need.
If you, or anyone else, decide to make your own(again, not super easy the first time), I'll attempt to describe it, in a "how to"
But to start, you'll need new cable, solder, flux, a 4 way tire iron(to be used as a solder pot), a bernzomatic hand torch(or similar), and new brass throttle cable ends(or wire terminal crimp connectors)
...Anyone visualize where I'm going with this??