Is the extra amount sent by the student's father, after he was informed of the reduced amount you charge your relatives, considered permissible (halal) or forbidden (haram), especially since you are in need of the money and he is well-off?

You can ask the sender of the money if he made a mistake in his estimation. If he sent it intentionally, there is no sin upon you in taking it, even if it exceeds the agreed-upon amount. If he made a mistake and requests the excess, return it to him. If he waives it, then take it.