NB: Psalms 42-43 can be regarded as one song. Perhaps Psalm 43 was composed as a later addition to Psalm 42 in the same way that composers might write an extra verse for a pre-existing song. The omission of the heading for Psalm 43 is probably for a similar reason to its ommission in Psalm 10, as Psalm 10 was clearly part of an irregular acrostic when combined with Psalm 9.
42:1
[HEADING]
For the Chief Musician. A riddle. Of the sons of Korah.
[FIRST VERSE]
Just like the deer pants after the streams of water,
in the same way my soul pants after you, O God.
42:2
My soul thirsts for God, for the living God.
When will I go and appear before God?
42:3
My tears have been my bread day and night,
while they are saying to me all the day, “Where is your God?”
42:4
These things I remember, and I pour out my soul within me,
for I would pass by with the throng,
and leading them slowly in procession to the house of God,
with the sound of joy and praise, a multitude making pilgrimage.
[REFRAIN]
42:5
Why are you cast down, O my soul?
And why are you unsettled within me?
Hope in God, for I will yet praise him for the salvation of his face.
[SECOND VERSE]
42:6
O my God, my soul is cast down within me.
For this reason, I remember you from the land of Jordan,
and the Hermonites, from the hill Mizar.
42:7
Deep calls to deep at the sound of your waterfalls.
All your breakers and waves have gone over me.
42:8
Yet the LORD will command his steadfast love in the daytime,
and in the night his song will be with me,
my prayer to the God of my life.
42:9
I will say to God my rock, "Why have you forgotten me?
Why do I go about mourning because of the oppression of the enemy?"
42:10
As by a sword in my bones, my adversaries reproach me.
All the day they say to me, "Where is your God?"
[REFRAIN]
42:11Why are you cast down, O my soul?
And why are you unsettled within me?
Hope in God, for I will yet praise him, who is the salvation of my face, and my God.
[THIRD VERSE]
43:1
Deliver judgement for me, O God
and argue my case against an unfaithful nation.
From a man of deceit and wickedness, rescue me,
43:2
For you are God of my protection.
Why do you reject me?
Why do I walk about in the dark,
in the oppression of an enemy?
43:3
Send your light and truth.
Let them lead me.
Bring me to the mount of your holiness,
and to your dwellings.
43:4
Then I will come to the altar of God,
To God the joy of my celebration
And I will praise you on the lyre, God, my God.
[REFRAIN]
42:11
Why are you cast down, O my soul?
And why are you unsettled within me?
Hope in God, for I will yet praise him, who is the salvation of my face, and my God.
Regarding the setting of the Psalm 42, in verse 9, the psalmist mentions the "oppression of the enemy", and in verse 10 "the reproaches of the adversaries", so probably there is an enemy or enemies making the psalmist's life miserable.
The psalmist is emotionally disturbed for some reason. In verse 3, the tears of the psalmist have been the psalmist's bread continually, day and night. Pastorally, this can validate an emotionally disturbed person, that they are indeed trusting God in the midst of their emotional turmoil.
In verse 1, there is the image of a thirsty deer panting after water to drink. The psalmist desires God, so he thirsts for God (v. 2).