Psalm 51's central request is easy to flatten into a general plea for forgiveness, but its own language is more specific than that. Having confessed the wrong plainly, the psalm asks first to be washed and cleansed, then moves to a distinct further request, "create in me a clean heart, O God; and renew a right spirit within me," using the Hebrew verb bara, elsewhere reserved almost exclusively for God's own act of creation in Genesis. The choice of that word signals that the psalm is not asking merely for the wrongdoing to be set aside or excused, but for something closer to being remade, a new inward disposition rather than a cancelled debt.
The psalm makes the same point again near its close, in verses often read as reflecting on the limits of ritual atonement itself: "thou desirest not sacrifice; else would I give it: thou delightest not in burnt offering. The sacrifices of God are a broken spirit: a broken and a contrite heart, O God, thou wilt not despise." Read against the wider sacrificial system the Hebrew Bible otherwise assumes, the claim is a striking one, that the state of the petitioner's own heart, rather than any offering brought to the altar, is what God actually receives as sufficient. This is what has made Psalm 51 the text so many later Jewish and Christian traditions of repentance return to when distinguishing outward atonement from inward change, treating renewal of disposition as the request the prayer is actually making, pardon being only its precondition.