Covering and Layer

Covering

The word covering refers to anything that is put on top of or around something. It is usually used for protection against outside elements.
Ex: The covering on your boat has come off. You will have to brush the leaves out of the cabin before you put it back on.
Ex: There is a covering on almost everything in my refrigerator. This makes the food less susceptible to bacteria.
The word covering can also refer to something that gives more information or explains an attached communication. In this case, you can also use the word cover.
Ex: You should send a covering letter when you send your CV. Employers will know you better, and a lack of information will not be a stumbling block to hiring you.
Ex: Did you send a cover email when you forwarded those pictures? This will prevent them from being overlooked.

Layer

The word layer means any substance or material that is spread or laid on top of something else.
Ex: After the eruption of the volcano, there was a layer of ash and dust across the town. It was so thick that you could not see the street.
Ex: Making seven layer cookies is easy. You just spread coconut, butterscotch chips, chocolate chips, and crushed graham crackers on top of each other and bake.
The word layer can also refer to the way people dress, especially during the cold winter months. It means that you are wearing more than one piece of clothing on top of another.
Ex: I envy you for being so warm blooded. I am always so cold that I dress in layers. Sometimes, I even wear three shirts!
Ex: My mother told me to add an extra layer. The weather report said to expect sleet, so she knew I should wear a jacket.