If a woman performs ghusl to purify herself from a second menstruation, does that ghusl suffice for the first menstruation from which she did not purify herself properly?

It is sufficient for the entire body to be covered with water during ghusl (ritual bath) through strong presumption; certainty is not required. Doubt about complete coverage after ghusl has no effect. So, if what you are referring to is merely a doubt, then it is of no consequence and nothing is required of you. However, if you are certain that your first ghusl was incomplete, then the ghusl for the second menstrual period achieves purity, and you must repeat the prayers you performed without a complete ghusl.