Ok, here's my take on it. The bird has a chunk of white feathers which, because of the lighting and nature of the features, appear as a fairly homogenous white area. The rest of the image is great, it's just this white patch. So, you could reduce the white on that area (or the image as a whole, but then suffer broader consequences), but if you did that, you'd end up with an image which isn't "real" or reflective of what you saw.
However, a patch of homogenous white just looks wrong to us - we're thinking "blown highlights" even if it's not. So, we need to introduce contrast and really define the subtle shade differences just within that patch.
I've used my limited PS skills to create a luminosity mask (actually something I've called a +3 whites mask, which is basically just a mask which half selects the brightest white and almost nothing else). I've then applied a simple contrast/brightness adjustment layer using that mask. I've set contrast +75 and brightness -75, which actually makes the brightest whites brighter but the rest of the bright whites greyer. What this does is create some differential within what was a previously homogenous white patch.
You might not like it, but I think it looks more "real" and still has clear white flecks but more grey and doesn't, IMO, look blown out. Note: because I've been editing the jpg it's suffered a bit of posterisation.