Is it permissible to marry a divorced woman secretly, with an agreement for a weekly meeting and the payment of the dowry and maintenance, and is this contract considered legitimate?

If the marriage is conducted with an offer and acceptance, the guardian's permission, and the presence of witnesses, then it is valid. Its validity is not affected by not publicizing it, because publicizing is recommended, not obligatory. There is no harm if a woman waives some of her rights.