To take "a" bite out of something often refers to literally biting and removing a piece of it.
When I wasn't looking she took a bite out of my hamburger.
One can figuratively take "a" bite out of something too, though this is less common.
McGruff the crime dog wants you to help him take a bite out of crime.
In this case, McGruff wants to reduce the number of crimes that are committed. In both cases we are reducing the quantity of something by taking a bite out of it.
To take "the" bite of something is almost always figurative. Jellyfish stings can feel like something is biting you, so you can "take the bite out" of them in various ways, including a mix of seawater and baking soda, which makes them less painful.
You can use "take the bite out" in many different contexts, with anything that feels like a sharp, biting pain -- physical, emotional, whatever.
In really cold weather like this you have to wear a warm hat to take the bite out of the freezing wind.
Actually it doesn't even have to be painful, just any strong sensation like an unpleasant odor or taste.
This curry is pretty spicy, but if you eat it with some of the yogurt sauce that'll take the bite out of it.
If we were to take "the" bite out of crime, we would be mitigating the negative effects of crime without necessarily reducing the number of crimes committed.
Synonyms for "take the bite out of"