When you score Nisei MK II card, do the agenda counters required to score it carry over to the scoring area?

Thus giving you 5 agenda counters to use the triggered ability:

"Hosted agenda counter : End the run."

(this question goes for other cards that have "Hosted agenda counter:" cost abilities)

Don't confuse advancement tokens with agenda counters. You put advancement tokens on the agenda to advance it so you'll be able to score it. Once you score it, those return to the token bank and the agenda gets one of the generic tokens that in this case represents an agenda counter.

So what does," hosted agenda counter: end the run" mean? You can end all runs as long as you have an agenda counter on it? That doesn't make sense!

JosephCurwen said:

Once you have scored the agenda you can remove the agenda counter from it to end a run. Since you put only one agenda counter on the agenda when you score it, you can use this action only once. 

JosephCurwen said:

It's a paid ability, recognizable by the template structure "<cost>: <effect>". You pay the hosted agenda counter to use the ability.