Final answer:
The priority is to administer oxygen via face mask to improve the patient's low oxygen saturation; assessing for a possible pulmonary embolism is also necessary once oxygen levels are stable.
Step-by-step explanation:
The priority nursing intervention for a female client reporting shortness of breath and lung pain, and showing signs of hypoxemia (SaO₂ 86%), is to administer oxygen via face mask. Oxygen therapy is crucial to improve the patient's oxygen saturation and alleviate hypoxemia, which can be life-threatening if not promptly addressed. As she has a background of smoking and recently started taking birth control pills, the risk of a pulmonary embolism is increased, but the immediate concern is to address the hypoxemia. It is essential, however, to assess for indications of pulmonary embolism once the patient's oxygen levels are stabilized, as this is a potential underlying cause of her symptoms and would require emergency treatment.