Is revealing hair for the purpose of travel visa procedures considered a necessity, and does the refusal persist even if it leads to the annulment of the marriage contract?

It is not permissible to photograph a woman with some of her hair uncovered unless there is a necessity or an urgent need that is considered equivalent to a necessity, such as photos for official documents (ID, passport, marriage contracts) if the authorities insist on it. If the questioner needs to travel, and not traveling would lead to harm, and she finds no way to obtain a visa except by uncovering some hair in the photo, then there is no blame upon her in that.