Can you put caustic soda in a car battery?

All of the below assume you are talking about a regular wet lead-acid battery. If you are talking about other types, please ignore!

Ok, back to the question: Well you can, a TINY pinch can help dissolve sulphation when the battery is next charged and warms up. However a SMALL pinch only (it will neutralise some of the acid as well as hopefully removing some of the sulphation, converting it to sodium sulphate).

EDTA or epsom Salts tend to be worth considering if you are going to put something in your battery. With both, dissolve in a small amount of water around a teaspoon per cell (for typical car battery). Do NOT just plonk in the cell as epson salts especially do not dissolve well in acid - but dissolve ok in water - especially if hot.

If you google for "battery desulphation" you will find other ideas if that is your problem - if you do not have a sulphated battery, then I wonder why you want to add caustic soda.