How do I deal with my ex-husband, who married another woman after I divorced him twice, and we remarried again after an illicit relationship, knowing that we have children and my family is unaware of the details?

You must repent to Allah for the forbidden relationship that occurred. Repentance involves abandoning the sin, regretting it, and resolving never to return to it, while maintaining discretion, avoiding public disclosure, and increasing good deeds. As for dealing with the ex-husband after the 'iddah (waiting period) has expired, it is permissible if there is a need and if temptation is secure, with seriousness, modesty, without privacy (khalwah), and without softening the speech. If temptation is not secure, then it is obligatory to avoid speaking with him and to suffice by appointing your maharim (unmarriageable relatives) or others to demand child support.