Deliver me out of the mire - Out of my troubles and calamities. See Psalms 69:1-2.
And let me not sink - As in, mire. Let me not be overwhelmed by my sorrows.
Let me be delivered from them that hate me - All my enemies. Let me be saved from their machinations and devices.
And out of the deep waters - See Psalms 69:1-2. From my troubles.