I already answered in detail a very similar variant (I know it's hard keeping up, it's hard for me too).
The difficulty is how to know in poly time that N accepts w, right?
The point is to use an algorithm that uses polynomial time and polynomial space as well (not constant space).
The idea is simple:
1. rid epsilon transitions it poly time (the procedure learned in class does that)
2. not use N to read w character by character, at each step update the set of states you can be at after reading w_1….w_i.
This can be done in poly time - at each step go over all states that currently you can be in, go over the transition function to compute all states you can reach given w_i, and update the set of states. so you need something like |w|x|Q|x|Q| to do that and that's poly time. you take advantage of the fact that in a TM you can keep a list of states at any point.