This is the thing that always rubs me wrong about these systems right here. The construction of a hard or soft move as the means of introducing a conflict and giving the PCs something to do is intuitive and makes sense here. I can see the value in introducing either one of those two scenarios, "go frame someone" or "go become infamous" as interesting problems/scenarios that PCs want to engage with. What I find baffling is that the system makes the resolution of all gameplay into yet more such moves, when I mostly feel like I'm done with that now, and ready to go see how resolving the situation turns out. I find myself wishing there was some other system that was used after a conflict was introduced, where I could do the standard game thing (minimize bad outcomes, maximize positive ones, determine the most efficient/safest/highest reward play to get to whatever goal we just established), but the loop just keeps iterating instead, which feels exhausting and unrewarding.
There
is a way to do it, through PC action. For instance, the PCs could decide that they don't want to be scapegoats and tell the captain of the mayor's duplicity. Or they could threaten the mayor into giving the help now. Or they could decide it's not worth it and turn the mayor's offer down. Or they could leave town and go somewhere else. Or they could make a counteroffer. And at this point, the GM has to then decide what move to use in reaction to that. Maybe you could have the mayor offer a different deal or accept the counteroffer (which may intrigue the mayor and lead to new adventure opportunities), or tell the PCs that they're no longer welcome in town and to leave immediately under threat of arrest (which may cause a loss of reputation--which is tracked in Root--in the area and other complications).
And, of course, that example only one potential move the GM can make. The PCs go to the mayor for help. One move is "Show signs of an impending threat." The mayor can give the PCs the help they requested and
also tell the PCs that there's a danger in the woods that seems to be getting closer and more violent. Alternatively, the mayor could use the move "Show them what a faction thinks of them," meaning that the mayor may be glad to help them (because he supports a faction the PC is well-known in) or may immediately call the guards (because he's against that faction); as a note, in Root, it doesn't seem to be
too hard to tell what faction someone is in--it's a bit species-related, and anyway, simply observing someone for a time would also provide the information. Or, while the PCs are asking the mayor for help, you could use the "Put them in a spot" move and have someone try to assassinate the mayor!
So basically, it's all the same options as in D&D and every other game, minus the "nothing happens" option.