How should you treat a puncture wound to the heal/palm lower part of the hand?

Attentively: This can be a very dangerous puncture wound.The problem is the blood supply and lymph drainage.If it does not look and feel good in a short time you must seek care asap.

Local hygiene: Wash with soap and water. Apply antibiotic ointment. Cover with dressing. Don't let it dry out. Check your tetanus status. If no better within days see a doctor immediately.

Observation: Unfortunately, puncture wounds tend to close vary rapidly so soap, water and topical treatments are essentially worthless. The problem is that the puncture pushes foreign material deep into the tissues and so not much can be done unless the problem materializes additional trauma from cleansing may worsen things. Make sure tetanus is ok and then watch carefully. Get to a doc for increased pain.