What is the purpose of the Article 4? Whether the Blessed Virgin should be called the Mother of God?

What Aquinas is trying to do is to justify the elevated position of the Virgin Mary in orthodox Christian theology, using carefully-reasoned logical argument to defend her unique status. As Christian orthodoxy held that Jesus Christ was both fully man and fully God, the question inevitably arose as to the precise status of his mother. Aquinas, using the scholastic method, deals with a number of objections to describing Mary as the Mother of God. The most powerful of these is the argument that the divine nature of Christ did not derive from his mother, hence she cannot be called the Mother of God. Although Jesus is the God-Man, we cannot say the same thing in relation to Mary.
Nevertheless, Mary enjoys the special status of theotokos, which means "bearer of God" in Greek. What this means is that Mary was indeed fully human, but at the same time graced by God in order to make her the fitting mother of his only begotten Son. In this sense, Mary is not just the Mother of God, but the highest of created beings, enjoying as she does a unique relationship to God the Son.
Aquinas draws upon this insight of the ancient Church fathers to argue that "God" in the expression "Mother of God" refers not to the Godhead, God the Father, but to God incarnate in the person of the Son, a person of both divine and human nature.