Psalm 142 is a prayer for help.  David freely voices his complaints and troubles and seeks refuge in the One Who cares for his soul, Who is a refuge and a portion.

Psalm 143 is a prayer for guidance when overwhelmed by circumstances. "Teach me the way in which I should walk; for to Thee I lift up my soul." David prays.( v8b)  "Teach me to do Thy will." (v10).  David puts all of himself in the Hands of the Lord, his God.

Psalm 144 is a prayer for help and also for prosperity.  In this prayer he proclaims the Lord to be:
His rock.
His trainer.
His stronghold and deliverer.
His shield and refuge.

David is amazed at the fact that God is mindful of man.  Of him. But, convinced of that fact, he goes on to ask God to do extraordinary things:
"Bow Thy heavens, O Lord, and come down." (v5)
"Touch the mountains, that they may smoke." (v 5)
"Flash forth lightning and scatter them; Send out Thine arrows and confuse them." (v6)
"Stretch forth Thy hand from on high, rescue me and deliver me out of great waters." (v 7).
David goes on to ask the Lord to bless the children, bless their garners, bless their flocks and their cattle. ( v12-14)

Big requests! What faith David displays in his bold requests.  What an example to us.... we need to recognize how great our God is, how intimately He knows us and thinks of us, and how big our prayers can be when we put our faith in Him!

These psalms ( along with all the rest) are such great examples and lessons on how to pray:
We need to pray for personal purity and purpose. (Psalm 141:1-5)
We need to pray for God's action against the wickedness in our world. ( Psalm 141:6-10)
We need to pray for God's help when we are in trouble or when we are overwhelmed. ( Psalm 142)
We need to pray for God's specific guidance for each day and in all of life. ( Psalm 143)
We need to pray, committing ourselves to the One who is our Rock, our Deliverer, our Shield, and our Refuge. We need to be mindful of the One who is mindful of us... 
We need to learn to ask Him for the big things as well as the small... expanding our faith, our trust, our complete dependance on Him for all we need and all our heart's desires.   ( Psalm 144)

Father in heaven, we come before You and bless Your Name.  You are our Rock, our Everything.  You are amazing and holy and perfect in every way!  You have made us and loved us.  You have made a way for us to come before You, to know You, and to bring every need before You.  We come in the Name of the Lord, Jesus Christ, who is our Redeemer and Mediator.  We come to ask for all these things that we need... for pure hearts and cleans hands; for help; for guidance; for righteousness and for deliverance. We come to ask for some big things... that You would open up the heavens and come down!  That You would set fire to mountains of hatred, mountains of false religion, and mountains of suffering.  We ask that You would stop those who love evil and do wicked deeds.  Those who kill unborn babies, and those who preach immorality.  Those who stir up dissent and love to disrupt the peace.  May they fall into their own snares.  May they repent of their sin.  May they find the One who alone can bring Light to their darkness.  Do mighty works of Grace that will turn the hearts of the lost to the Savior, Lord.  For only You can do such things.  You are the Lord and we bless Your Name. We give You thanks for all that You have done. Amen