Is it permissible to give charity to a single woman with a very low income, knowing that the giver is aware of her unwillingness to disclose her situation and that there is no kinship between them, or is this considered a whisper from Satan?

Giving charity to such a woman is permissible, provided one avoids the means that lead to evil and temptation. The money can be delivered to her by a trustworthy woman or one of the giver's mahrams, without informing her of his identity, in order to prevent temptation.