The click sounds like you are describing the a.starter or b. sellonoid , probably from a dead battery, i am not sure how those systems work, by that i mean if it has a fuel injector, or carb. But the part about you lollygagging the throttle and it wanting to bog down could be any of the following or a combination of two or more together. 1. Your air to fuel mixture is not right, causing to much air in the motor when you gag it, a simple screw adjustment could fix that. Or 2. it might, I say might in a longshot manner) could be your battery is either dead or dying, low on distilled water only if you refill (check it, then charge it, if low ) check to see if it turns over.)new battery would e the est option. But the other should work for now, if thats the issue. Not enough volts can cause bogging down as well as click, thats the only reason I think it possibly could that. 3. Cut or old, bad plug wire that is allowing the spark to jump with a load on it. 4.) timing could be off. You might be hearing the governor, if you lollygag a moped that has governor on it, that's what it is supposed to be doing. 5.) could be a cracked,loose of bad fuel line, or vacuum line, this is at the top of the list of things to check once you get it to crack. When you gag it, the sudden increase of air and fuel pressure causes the line to collapse, of draw air through the crack. Does the moped idle nice and smooth? Accelerate smooth? Maintain steady smooth ride at one speed? 6) clogged fuel filter. 7) cracked or bad spark plug 8) how fast is it supposed to go? If it does near that speed just fine, i would guess it might be the governor kicking in when you try and exceed its designed limitations. I think you have lose connection to the starter or starter selinoid gone bad, and the governor is kicking in. Sounds like a wire wiggled loose causing it to die, look for a wire to tighten up.