What is the ruling of the Sharia on the continuation of a marriage with a husband who abandons his wife and children for many days, and does not provide for them regularly, knowing that the marriage contract was customary, with witnesses and a dowry?

This marriage is invalid due to the absence of a guardian (wali) and witnesses. The guardian could have been present or appointed a representative to act on his behalf; otherwise, the ruler (sultan) is the guardian. A woman cannot appoint a representative unless there is no guardian or ruler. As for witnesses, it is a condition that two witnesses be present, neither of whom is the guardian or the husband. Therefore, you must immediately separate from this man and repent, and the marriage contract can be renewed in a lawful manner.