The carb has a wee screw you need to use a screwdriver on and a thumb screw. Wee one is 1.5 turns out and the thumb screw I use for adjusting the idle?

Bike was out today and was a bit better. If you caught it fouling you could actually rev it out to clean it. Before it would be the first sign of the plug about to crap out.

Is the wee screw nearer the engine, or air filter ?

Wee screw is nearest the engine.

Bike has been sitting since last outing. Went to start it today as the laddie was wanting to go out tomorrow and it won't start.

It acted as if it was running out of fuel like it usually does before the plug dies. I stuck it back in the garage in the huff but will need to get it running later.

I'll clean everything again and fit another new spark plug. This is really ruining the enjoyment of the wee bike and I'd love to get it sorted.

I have a Beta 80 with the same issue, killing a spark plug after a few hours riding with lots of black carbon deposits. I just spoke to Beta UK, they suggested using a BP5ES spark plug. Apparently the higher the number, the better the plug for hotter climates. If you're riding in the UK they suggested a lower number like 5 (rather than the 8 I was using) would be better for our weather temperatures. I hope that helps. 

Seems like establishing what’s rich and lean will help too. You might have an overly rich pilot jet if air screw (small flat head brass one) is set right. I’d try the hotter 5 plug like has been suggested. This alone may cure it. If not, open the air screw (not idle) to two turns out. This allows more air to come in at idle. What happens? Does it die as you open the throttle with a lean bog? You can open the AS to around 2.5. Just for reference these are what each carb part is doing if you don’t know. Marking the grip and the cable body on the throttle will help you ID which bit is wrong.

Pilot jet. Idle. Along with AS. 

Needle diameter. 1/8th throttle. 

Needle clip/taper. 1/4 to 3/4 throttle. 

Main jet. 3/4 to full throttle. 

You can’t cure an idle/initial throttle issue by changing main jets and clips. Clip might help a bit but it’s too far into the throttle range really. 

Winter settings use more fuel, summer is leaner more often than not. 

I’d be surprised if this isn’t just plug heat range and pilot/air screw related. 

Just a guess, but I do own 2 of these Beta's. The exhaust and silencers fill up with oil as they don't run hot enough to burn it away.It's not helped by riders constantly "blipping" the throttle and stop start riding which is common in the younger age group.

I run about 50ml of 2t to 4 L which is about 80.1which helps .

While I'm on leaking petrol taps are common (Always switch them off) they Make a nice job of filling the crank with fuel and this really adds to the problem .

Cheers the noo.