Hi Anthony - welcome to WAB.
Such a shame to lose youngsters like that
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I'm no expert, but like the others I don't feed peanuts this time of year. I think Greenfinches are mainly seed eaters, but will have a go at other things - especially babies, who I imagine are curious like other babies are. Peanuts get stuck in little throats, hence the general advice not to feed peanuts during the breeding season.
The only other thing that came to mind for me was the inclement weather bit. You don't say how you put out the peanuts, but if you are going to do that you need to replace them frequently, because if they get damp and aren't replaced, they can go mouldy, and I'm not sure if I'm absolutely 100% right on this, but this is toxic to birds, although I don't know how quickly it might kill them
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I hope that might help you.
Thea