The verses above are from a Psalm written after Nathan the prophet confronted David for the adultery he committed with Bathsheba...
In verse 10 you see David's plea to have God create a clean heart and renew a right spirit within him. There is nothing so filthy, impure, or as loathsome as the human heart prior to being converted. Our hearts are so corrupted and vile that mending them will not work - they have to be made altogether new and only God himself can do so. It has been said that a clean heart has been purified from guilt and cleansed from filth and it is seen here to be David's great desire.
In verses 11 and 12 we see David pleading with God to not reject him, abandon him, leave him in sin and sorrow, nor take his Holy Spirit from him. Having once tasted the joy of his salvation, David asks for this great joy that was racked by sin to be restored again. James Smith, the predecessor to Charles Spurgeon at New Park Street (1841-1850) states:
When the Spirit of God once takes up his residence as the Spirit of life, of love, and adoption, he never finally withdraws, or abandons the soul to ruin. But though he does not totally and finally withdraw, he withholds his influence, and leaves us pretty much to ourselves. He ceases to comfort us, or to strengthen us with might, according to his power in the inner man. He refuses to assist us in prayer, and we are left to toil alone. Then like Samson, we become weak, and are like other men; or like the church we cry out, "The Comforter that should relieve my soul, is far from me!"
When the Spirit is withdrawn from us, all our graces wither, our evidences become obscured, our joy dies out, and our duties become a task. Then we become light and frivolous, or gloomy and morose; our hearts are hardened, our understanding is darkened, and all within us appears carnal or cold. The greatest loss we can sustain, is the presence and operation of the holy Comforter in the heart. Then very often doubts, fears, and misgivings spring up, unbelief works, and a sense of condemnation is felt in the conscience. We turn everything against ourselves, refuse to be comforted, and begin to neglect duties, because we have no enjoyment in them. Past experience looks like delusion, the hard heart refuses to weep, and the mouth is often ready to speak against God. Sad, very sad, has been the experience of many professors, when the Spirit has left them for a time; well therefore may David pray, "Take not your Holy Spirit from me."
In verse 13 we see the desires of a grateful heart revealed. When we have had our hearts of stone turned by God to hearts of flesh and when by God's grace he renews a right spirit within us it will then be the heart's great desire to share the gospel to a lost and fallen world, to go tell it on the mountain top, and to yell it from the roof top. Sinners need to have God's truths shared with them...they need to be made aware of their sins, and they need to hear the good news of Jesus Christ. There is an old British proverb that says "reclaimed poachers make the best gamekeepers." My friend if God has created in you a clean heart and if he has given you the joy of his salvation then go...go and share what God has done and point sinners to Jesus Christ.