Is it permissible for a widow to marry someone she secretly loves without her son's knowledge, bearing in mind that the husband is already married and wishes to keep the matter secret from his family and children, and what are the conditions for the marriage to be valid?

If the marriage (nikah) is conducted with the guardian's (wali's) permission and in the presence of witnesses, it is valid. Announcing the marriage is recommended, not obligatory. If the guardian prevents you from marrying someone you approve of, you may refer the matter to the Sharia judge to arrange your marriage. However, marriage without the guardian's permission or without witnesses is invalid. As for your son who has reached maturity, you should hasten to marry him off if possible.