Some more observations based on 51.12-21 (ET 51.10-19) drawn from Hossfeld and Zenger's new Hermeneia commentary on the Psalms. See here for the previous post.
1. The petition "create in me a clean heart and renew a correct spirit" picks up the language of new creation (the same verb is used as in Genesis 1.1) and of prophetic promises to exiled Israel. Thus this is a prayer by which the individual seeks to locate themselves within the wider story of God's redemptive purposes for creation and thus the world. This is confirmed by the phrase אַל־תַּשְׁלִיכֵנִי מִלְּפָנֶיךָ in v13 which is also used to describe the banishment of Israel into exile. The technical phrase for this idea is 'individual eschatology' whereby the Psalmist claims for themselves the eschatological promises made to Israel. These promises of deliverance and their consequent fulfilment in the life and experience of the Psalmist constitute God's righteousness the צִדְקָתֶךָ of 51.16 - thus this is an example of the notion of 'righteousness' equating to God's saving power for Israel and the individual: this is the likely background to Paul's language of the 'righteousness of God δικαιοσύνη θεοῦ in Romans.
2. The relationship between individual and people, the Psalmist and Israel, and the interaction between their present situations (exile, sinfulness) and promised future (forgiveness, restoration, salvation) explains the presence of 51.20-21, which pray for the rebuilding of Zion. The verses are secondary, but the Zion theology expressed within them suggests a canonical reading whereby 'the renewal of Zion as central point of creation begins with the new ceation of its inhabitants.' (Hossfeld and Zenger, 23).
3. The phrase וְרוּחַ נְדִיבָה in verse 14, translated 'willing spirit' prepares the way for the imagery of living sacrifice that follows. The root is actually a technical term used to denote the 'freewill offering' (see Exod. 35.5, 21).
4. Psalm 51 may well lie behind the overall shape of the story of the prodigal son in Luke 15.11-32 as well as the specific wording of the son's confession in 15.18, 21. The structure of acknowledgment of guilt, confession before a merciful God/father, new creation and festive meal/sacrifice is broadly similar. The theology that we have begun to articulate in these comments also shapes the opening chapters of Romans where Psalm 51.6 is cited (in the LXX). This suggests that attempt to set an 'individual' reading of Romans (how can I find a gracious God?) over against a 'covenantal' reading (is God faithful to the covenant with Israel?) should not be overplayed. In this, the most individual of the Psalms, it is Israel's story being told, and indeed the covenantal promises of God to Israel are the basis for the personal expressions of faith and confidence to which this Psalm gives voice.