Is money sent from a married woman—with whom you communicated using a pseudonym and who learned of your dire financial situation—considered haram (forbidden) money?

You must hasten to repent to Allah by cutting off your relationship with this woman and other non-mahram women. Repentance involves abandoning the sin, regretting it, and resolving not to return to it. As for the money the woman sent you, if you clearly expressed your need to her and were truthful in that, and she gave you the money out of consideration for your need and not for an unlawful purpose, then it is permissible for you. However, if she gave it in exchange for an unlawful relationship, then it is forbidden for you, and you must dispose of it by spending it on public welfare and charitable causes.