Is it permissible to seek khulʿ from a second husband in order to return to the first husband to care for the children, especially given the possibility of the new wife neglecting them?

A woman is not permitted to seek a khul‘ (divorce at the wife’s instance, by returning the dower or other compensation) without a legitimate religious reason, because the Prophet, peace and blessings be upon him, described those who do so as hypocrites, meaning disobedient inwardly and obedient outwardly. Your desire to return to your first husband for the sake of the children does not justify your request for khul‘, because the children are now under the responsibility of your second husband, who must provide them with custody. You have forfeited your right to custody by marrying another man. If there is a suitable female for custody, she has a greater right to the two children.