Jesus gives a line of this psalm "that the Scripture may be fulfilled" (John 13:18). But the same psalm says, "I have sinned against you" (41:4). Is the whole psalm about Jesus?

John gives Jesus' words about one line. Jesus says, "that the Scripture may be fulfilled, ‘He who eats bread with me has lifted up his heel against me.’" (John 13:18), which is the second half of Psalm 41:9. He doesn't quote the rest of the psalm, and John doesn't say how its other verses bear on Jesus. The psalm says who prays 41:4: "I said, “LORD, have mercy on me! Heal me, for I have sinned against you.”" (Psalm 41:4). It is the psalmist's own confession, in a psalm where his enemies say an evil disease has afflicted him (Psalm 41:8). Jesus says why he told them beforehand: "From now on, I tell you before it happens, that when it happens, you may believe that I am he" (John 13:19). So John answers for one line and doesn't say whether the rest of the psalm speaks of Jesus. These notes go no further than John: they don't carry 41:4 over to Jesus, and they don't say more about the rest of the psalm than the New Testament says.