pastorbobnlnh 

I have experience servicing Helicomatic freewheels, but I've never actually ridden a bike equipped with one. I do have a few Helicomatic freewheels available for purchase and a fair supply of replacement sprockets.

With that said, I'd try using a better 9-speed SRAM or KMC chain, after cleaning the sprockets, derailleur jockey pulley/wheels, and chainrings. You don't mention if the shifting challenges are across all the sprockets or only on certain ones. If only on certain ones, there's a good chance the sprockets are worn and need replacing. In your two pictures of the freewheel, I see significant grime, grit, etc., and wear on the smallest sprocket.