A.W. Pink addresses the issue of despondency through the lens of David's struggles, emphasizing that even in times of deep sorrow, one must confront their troubles and seek the root cause of their despair. He encourages self-examination and the importance of not succumbing to despair but rather preaching hope to oneself by trusting in God's promises. Pink highlights that despondency often stems from unbelief and a rebellious will, and he reassures that God is a present help in times of trouble. The sermon concludes with a call to hope in God's mercy, power, faithfulness, and love, reminding believers that joy will come after trials.