The first clause of the verse simply says, Hear, Lord, what is right [צֶדֶק].
So the issue revolves around how to understand the meaning of what is right.
Commentators generally agree that it needs to be interpreted in light of the parallelism with the second clause, particularly the phrase my cry.
That is, the possessive pronoun my
(which appears in the Septuagint, the Greek translation of the Old Testament, my righteousness
) is implied, so that the term could mean my just cause
or my righteous plea
(NIV). In other words, the psalmist’s cry is that the Lord would hear and take seriously his just cause.1
1 Hear a just cause, O LORD; attend to my cry! Give ear to my prayer from lips free of deceit!