Couldn't find this anywhere, so I have to ask...

If a limited power is negated (like Quicksilver, Speed Demon) can the power be used again if the payment/condition can be met?

no, it has already been used for the turn. you negate a power on the chain. when you use that power you use it to put on the chain. so if its used once to put it on the chain, then it is negated, then you have already used it up for the turn

"Use this power only once per turn" means "generate this effect only once per turn".

Whether the effect is negated does change the fact that it was generated once already.