Q. Should the word head in this sentence have an s on it to agree with the multiple kids, or does this create a situation where each kid ends up with multiple heads? “The children put their hats on their heads.”

A. Add the s. Please note, however, that people don’t always talk that way; the construction that omits the s is common and accepted in many contexts. (Of course, if the children have a single head, they should have a single hat, pace Dr. Seuss.)