**Core Concept**
The question is testing the recognition of a common pediatric injury known as "Nursemaid's elbow" or radial head subluxation. This condition occurs when the radial head is partially dislocated from its normal anatomical position, often due to a sudden jerking motion of the forearm.

**Why the Correct Answer is Right**
Radial head subluxation typically affects young children, especially those between the ages of 2 and 4. The injury often occurs when a child is lifted by the hand or forearm, causing the radial head to become partially dislocated. The characteristic symptom is a sudden onset of elbow pain and refusal to move the affected arm. The correct management involves a specific maneuver to reduce the subluxation and restore normal anatomy.

**Why Each Wrong Option is Incorrect**
**Option A:** Applying a cold compress may provide temporary relief but does not address the underlying cause of the injury.
**Option B:** Administering analgesics or anti-inflammatory medications may help manage pain but does not treat the subluxation.
**Option C:** Immobilizing the elbow with a splint or sling may provide temporary support but does not facilitate reduction of the subluxation.

**Clinical Pearl / High-Yield Fact**
In pediatric patients, radial head subluxation is often diagnosed based on clinical presentation and can be reduced by the "supination" maneuver, where the child's forearm is gently rotated while applying gentle pressure to the elbow.

**Correct Answer:** C.