Dear K:

You asked:

In the following sentence, is the usage of the word “cook” correct or incorrect?

“She insisted on helping me cook.”

Must one write “She insisted on helping me with the cooking.”?

Either one is fine. Both are perfectly good English. In the first one “cook” is an infinitive acting as a direct object of the gerund, “helping.” In the second one “cooking” is a gerund acting as the object of the preposition “with.”