Fair Access to Legal Counsel Act of 2020 This bill requires a court to consider specified factors when determining whether to provide counsel for a civil litigant who cannot afford counsel. In addition, the bill authorizes a court to appoint, rather than merely request, counsel for such a litigant. Under current law, a court may request counsel to represent a civil litigant who cannot afford counsel. The bill instead authorizes the court to appoint counsel upon request by the litigant. A court must inform civil litigants appearing without counsel of their right to make such a request. In considering whether to grant a civil litigant's request for counsel, a court generally must consider factors such as the complexity of the case, any mental or physical limitations faced by the litigant, and whether any opposing party is represented by counsel. The court may also appoint counsel for limited purposes, including for the purpose of advising the court as to whether a full appointment is warranted. The bill also eliminates the requirement for a court to dismiss a civil litigant's case if the litigant's allegation of poverty is untrue.