From the details provided trauma, swelling, inability to close fist and the image shown there is a high possibility of fracture and visit to ER is recommended.
Home treatment can be given in form of ice packs, elevation of hand above the level of heart by flexing the elbow joint, use of OTC available antiinflammatory such as NSAIDs (if he is not asthmatic or allergic to it) along with enzyme supplement such as bromelain, papain, trypsin are required to counter the swelling.
Encourage him to actively wriggle the fingers as much as possible.
Take your child to ER when he is willing for xray of the affected hand. Fracture of head of 5th ray is common in such scenario, depending upon fracture pattern it may require splintage or surgical intervention.
Feel free to discuss further.
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