Hare's set-up is as follows:
When you are considering whether to aid a needy child, you care about two things, your well-being and the well-being of the child. When both you and the child are better off in one state of affairs than another, then you prefer the one to the other. But when you must trade off your well-being against the child's how you make the trade-off depends on your relation to the child. When the child is physically near, psychologically salient to you, such that only you are in a position to save him, and in a rare relation to you (for short: near etc.), you place greater weight on the child's interests--"I can't have nearby children dying for want of help," you say. When the child is physically far, not salient to you, such that others are in a position to save him [but won't], and in a common relation to you (for short: far etc.), you place greater weight on your own interests--"I can't solve all the world's problems myself," you say.
Hare then contrasts a Near case where little Ned has a comparatively less happy life at stake, with a Far case where little Ned would suffer a comparatively worse death if not saved. By dominance reasoning (it being better for Ned and worse for no-one), Hare claims you must both (i) prefer the state of affairs of saving Ned in the Far case over saving Ned in the Near case; and (ii) prefer the state of affairs of leaving Ned to die in the Near case than leaving him to die in the Far case. So, given that you prefer saving Ned in the Near case over leaving Ned to die in the Near case, you must similarly prefer to save him in the Far case rather than letting him die in the Far case -- else your preferences across these four states of affairs will prove cyclical.
But is it true that saving Ned in the Far case (letting Ned die in the Near case) is no worse for you than saving Ned in the Near case (letting Ned die in the Far case)? Hare can stipulate that the immediate financial costs are no different between Near and Far. But we may expect there to be significant differences in the psychological costs. Letting a near etc. child die would expectably be a cause of guilt and (if others learn of it) shame. Saving a far etc. child may place psychological (consistency) pressure on you to save others of the many far etc. children that remain, thus leading to greater long-term financial costs. So it's not so clear that dominance reasoning forces us to have the preferences that Hare claims it does.
Could these other differences just be stipulated away too? Suppose that your memory of your choice will be immediately wiped away, and that nobody else will ever learn of your role in it. In those (weird) circumstances I think Hare's argument goes through (though I could see someone arguing that having performed a shameful act would be a stain on your life, even if nobody actually knows of this). But given that we are never in those weird circumstances, the significance of this result seems limited. Update: As Hare has since pointed out to me, this objection isn't so dire after all. All he needs to restore the dominance argument is to be able to add additional "costs" sufficient to outweigh the psychological costs I've flagged, without changing our judgment that we're required to prefer saving in the Near case over letting die in the Near case. And it seems plausible that this could be done...