At 6 months old, a goldfinch should definitely be done with its first major molt, so that lingering bald patch around the neck and head isn't typical. It could be down to a dietary deficiency during the moult (like a lack of proper amino acids, proteins, or specific vitamins), stress, or potentially a mild case of feather mites or skin irritation if the new feathers are struggling to push through. Checking their diet and maybe supplementing with some high-quality seeds, soft food, or multivitamin drops geared toward moulting usually helps, but if the skin looks red or scaly, it's worth keeping an eye out for parasites.
