That information about how to check the Honda dipstick is correct of course. Also, as Rob said, if the sump is grossly overfilled it could cause substantial amounts of oil to pass through the disk valve into the valve compartment. If that happens, some of it will get into the breather hose, and the normal gas pulses in the hose will blow it into the carburetor's air intake. So, check the oil level before concluding that the ring gaps have to be checked.
Unfortunately Paul, to check the ring gaps you have to take the engine off the mower, remove the cylinder head and crankcase cover, then remove the connecting rod, complete with piston and rings. Then you remove the piston rings from the piston and put them one at a time, into the cylinder bore, 1 cm down from the top, ensuring they are square to the bore by pushing them into that position with the top of the piston. Then you use feeler gauges to measure the gap in the piston ring. When you get to that point (and you should only go there if the oil level is correct in the sump, of course) we can talk about how worn the rings are, and what to do about it.
If the disk valve were stuck closed, high pressures would eventually (depending on how worn the piston rings were) develop in the crankcase, and most likely oil would be forced past one or more of the crankcase gaskets, possibly blowing the gasket while doing so. If the valve were stuck open, there would be more oil than normal in the PCV valve compartment all the time, and this would result in there always being some oil in the breather hose. Your symptoms seem to be a decent fit for a stuck-open PCV valve, which is why I suggested you check for that.
The amount of wear on your valve disk is normal for a Honda that has just about worn out its piston rings.
