Is it permissible to give the value of feeding ten poor persons to one woman, knowing that she is a mother of six individuals, and there remain four poor persons?

If someone vows obedience to Allah but is unable to fulfill it, they must offer an expiation for breaking an oath (kaffarat yamin). So, if you vowed obedience but were unable to fulfill it due to incapacity, then you only owe the expiation for breaking an oath. There is no harm in considering the poor woman's situation and that she is a mother to six needy children when giving her zakat, provided she is not wealthy. As for paying the monetary value instead of feeding the poor, the majority of scholars hold that it is not valid, contrary to the Hanafis. If we assume the validity of what you did, you still need to feed four more needy people. You can delegate a trustworthy person or a reliable charity to pay the expiation to those deserving, even if they are outside your country.