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Sprinkle baking soda all over thsink surface and leave it overnight. Then next day remove using using brush and warm water. This will surely help.
You might have some food particle stuck in the drain trap or the caulking is rotten with mold, I would try to snake down the drain or use chemical drop cleaner and see if it helps.
I am going to try pouring bleach down the drain. I checked underneath the sink and there is no leaking. Thanks for the tip.