All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
The Hadeeth which includes this supplication: “O Allah, I appeal to You for the weakness in my strength and my limited power and the treatment of contempt and humiliation from people. To You, the Most Merciful of all the merciful ones, You are the Lord of the oppressed, and You are my Lord. Under whose care are You leaving me to? To an enemy oppressing me? Or to a friend You have given control of my affair? If there is no anger from You on me I will forever be content. However, Your blessing is vastly important for me I seek refuge with the glory of Your Light which the heavens and earth are lit form, Your anger will not befall on me, nor Your displeasure descends on me. To You is the supplication until You are pleased and there is no control or power except by You.” was reported by At-Tabaraani on the authority of ‘Abdullaah ibn Ja‘far but some scholars classified its chain of narrators as Dha‘eef [weak]. However, it is permissible to supplicate with such supplication because it is permissible to supplicate with anything that has a correct meaning and which does not have an Islamic prohibition even if there is no authentic Hadeeth about it.
There is no doubt that this supplication which you are inquiring about has good expressions and good meanings, good humility and humbleness in front of Allaah, the All-Mighty, the Oft-Forgiving.
In addition to this, the majority of the scholars are of the view that it is permissible to act according to a Dha‘eef Hadeeth provided the following conditions are met:
1- It should not be extremely Dha‘eef.
2- It should fall under a principle that is acted upon.
3- One should not believe that it is authentic when he acts according to it.
For more benefit, please refer to Fatwa 166199.
Allaah Knows best.