Is it permissible to continue speaking secretly with a friend after one's mother has been informed not to speak with him, out of fear of angering her, and is this considered a promise or a covenant?

If the friend is not a bad influence, then there is no harm in speaking to him without the mother's knowledge. What benefits a person and does not harm the parents does not require their obedience in prohibiting or permitting it. Your promise to her not to speak to the friend is not binding, but you should not promise what you do not intend to fulfill, for that is a lie. Instead, alluding and equivocating can be used. And righteousness towards the mother is among the best acts of devotion and the most beloved deeds to Allah.