Final answer:
The medication anticipated to be administered next to the lethargic, pale infant with significant circulatory compromise is epinephrine, which is the first-line medication in pediatric advanced life support for bradycardia with poor perfusion.
Step-by-step explanation:
In the scenario described, a 10-month-old infant with a heart rate (HR) of 38 bpm and a blood pressure (BP) of 58/38 mmHg showing signs of significant circulatory compromise has already been initiated on ventilation with bag-mask and chest compressions. With intravenous (IV) or intraosseous (IO) access established, the next medication anticipated would typically be epinephrine. Epinephrine is the first-line medication in pediatric advanced life support (PALS) for patients with bradycardia with poor perfusion despite adequate oxygenation and ventilation, which seems to be the case with the described weak pulses and prolonged capillary refill.