I don't think that we'll improve or finish fourth mate. I don't trust Moyes at all.
I also think though, that he should get a 'fair' chance. Not because he has done anything to deserve that - he has pretty much done the worst possible job since he came here - but because 1) the club decided to hire him, 2) I doubt that an another manager will improve things to save the season, 3) the club likely haven't been looking for a replacement so if we now do that on panic, likely it will be Curbishley or Fat Sam or another Premier League proven manager who will make things somehow even worse, 4) there is a slim - very slim - chance that maybe somehow things will get better.
Saying that, I wouldn't be sad if when I wake up in the morning, I read that Moyes has been sacked. Would be pretty happy if the replacement is Bielsa (likely the best unemployed manager at the moment) or Hiddink.
As of now, I hope that the club is active looking for solutions on the summer. Preferrably for foreign modern managers. Maybe better than way then panicking now.