A plea for forgiveness can rely on human merit, but a deeper appeal rests entirely on the essence of God. King David asks God to pardon him for the sake of His name, an approach designed to awaken divine mercy [מצודת דוד, מלבי״ם]. Because God is known as being good, He grants grace to people far beyond what strict justice demands [רד״ק, מאירי]. Furthermore, seeking a pardon for the sake of God's name serves a broader purpose. It makes His forgiving nature known to the entire world [אבן עזרא].
In his prayer, David focuses on one specific failure. This points directly to his most severe act, the incident with Bathsheba. Even though he had already sought a general pardon for his past actions, he returns to request a special, private forgiveness for this specific wrong [רד״ק, מאירי]. The type of wrongdoing he describes represents a sin committed through a twisting of the mind. This makes it more severe than a simple accident or an act of open rebellion. Because of its severity, David asks for the highest and most absolute form of erasing sins, which is a complete forgiveness [מלבי״ם].
David acknowledges the immense size of his failure, a declaration understood in three complementary ways. First, it serves as a plea for God to forgive the wrong even though it is so vast [אבן עזרא, מצודת דוד, מאירי]. Second, the statement acts as a deep confession. Taking a failure lightly prevents a true pardon, so David emphasizes the heavy reality of his actions. By fully admitting with a whole heart that his wrongdoing is massive, he becomes worthy of atonement [אלשיך]. Finally, the vastness of the failure is matched by the greatness of God Himself. David asks for a pardon for the sake of God's great name, expressing that it is entirely fitting for the Great Master to forgive a great sin [רש״י, אלשיך].