A Missouri commission charged with determining the fate of the state’s last abortion clinic will hear the case in August, weeks after the expiration, set for Friday evening, of the injunction that has kept the clinic operating.

Lawyers representing the Reproductive Health Services of Planned Parenthood of the St. Louis Region filed a motion to the state’s Administrative Hearing Commission Tuesday asking it to allow the clinic to continue operating as an abortion provider, as the commission considers the case.

After...