Answered by Sayyidi al-Ḥabīb `Umar bin Hafīẓ (may Allāh preserve him and benefit us by him).
How should we deal with mental illness amongst our youth? Is there a light dhikr that people can read that can ease their state?
In general any dhikr or prayer upon the Prophet (may Allah bless him and grant him peace) is beneficial in this regard. The following adhkar can be read specifically:
Read 100 times a day:
رَبِّ اشْرَحْ لي صَدْرِي و يَسِّرْ لي أَمْرِي
Rabbi-shraḥ lī ṣadrī wa yassir lī amrī
(“My Lord bring tranquillity to my heart and give me ease in my affair.” Ta Ha 20:25-26)
Read 70-450 times a day according to the need:
حَسْبُنَا اللَّهُ وَنِعْمَ الْوَكِيل
Hasbunallāhu wa n`am al-wakīl
(“Allah is enough for us and He is the best One in whom we place our trust.” Al `Imrān 3.173)
Read 11-100 times a day:
و أُفَوِّضُ أَمْرِي إلى الله إِنَّ الله بَصِيرٌ بِالعِباد
wa ufawwiḍu amrī ila’llāh inna’llāha baṣirun bi’l`ibād
(“I resign my affair to Allah, Allah is fully aware of (His) slaves.” Ghāfir 40.44)
Read Surat al-Inshirāḥ (94) and Ayat al-Kursī after every prayer.
Surat al-Ikhlāṣ, al-Falaq and al-Nās are also effective in warding off different types of harm.