Is it permissible for a woman to travel for three hours without a mahram, accompanied by her seven-year-old child, for the purpose of necessity, earning a livelihood, visiting relatives, and seeking knowledge?

A woman is not permitted to travel unless accompanied by her husband or a mahram (definition). A seven-year-old child is not considered a mahram for her, as a mahram must be an adult. This is because the purpose of a mahram is to protect the woman, and this cannot be achieved except by the presence of a mature, discerning adult.