What is the ruling on a man's relationship with a married woman, where he fears for her and her children in her husband's absence, with emphasis on the relationship being devoid of any transgressions other than respect?

If the woman mentioned is not one of your mahram, then she is a stranger (ajnabiyyah), and it is not permissible for you to be alone with her. This is based on the Prophet's (peace and blessings of Allah be upon him) prohibition of entering upon a woman whose husband is absent. Establishing a relationship with her is one of the steps of Satan, leading to indecency and evil. It is obligatory to be cautious and avoid any closeness to her that leads to what is forbidden. If there is a need to speak with her, there is no harm in that, provided it is not in seclusion and subject to certain conditions.