Overview

The main aim of this study is to see how well the medicine zasocitinib works, how safe it is, and how children and teenagers aged 4 to under 18 with moderate-to-severe plaque psoriasis respond to it.

The study will be done in 2 parts: Part A will include both children and teenagers, while part B will only include children.

At first, only teenagers who meet the study rules can participate in this study. Children may only start to participate once enough information has been collected from other studies with zasocitinib.

Participants in Part A will initially be assigned to receive either zasocitinib or placebo for the first 16 weeks of treatment, then all participants will receive zasocitinib through the end of the study. All participants in Part B will be assigned to receive treatment with zasocitinib throughout the study.

Participants will be in the study for up to 4 years and 2 months (217 weeks), including up to 35 days for the screening period, 208 weeks of treatment (Part A and Part B) and a 4-week safety follow-up period. During the study, participants will visit their study site multiple times.

Eligibility

Inclusion Criteria:

1. Participant has a diagnosis of chronic plaque psoriasis for greater than or equal to (\>=) 6 months prior to the screening visit.
2. Participant has stable plaque psoriasis defined as no significant flare or change in morphology (as assessed by the investigator) in psoriasis for \>=6 months before screening.
3. Participant has moderate-to-severe plaque psoriasis as defined by a Psoriasis Area and Severity Index (PASI) score \>=12 and a Static Physician's Global Assessment (sPGA) score \>=3 at screening and Day 1.
4. Participant has plaque psoriasis covering \>=10 percent (%) of total body surface area (BSA) at screening and Day 1.
5. Participant must be a candidate for phototherapy or systemic therapy.
6. Inclusion Criteria for Part A Cohort 1: The participant is male or female and aged 12 to less than (\<) 18 years, inclusive.
7. Inclusion Criteria for Part A Cohort 2 and for Part B: The participant is male or female and aged 4 to \<12 years, inclusive.
8. Inclusion Criteria for Part A Cohort 1: The participant must weigh \>=40 kilograms (kg) at the time of screening.

Exclusion Criteria:

1. Participant has evidence of nonplaque psoriasis (erythrodermic, pustular, predominantly guttate psoriasis, predominantly inverse, or drug-induced psoriasis). If a participant meets criteria for inclusion based on typical plaque psoriasis presentation, a limited amount of inverse psoriasis is not exclusionary.
2. Participant requires systemic treatment, other than nonsteroidal anti-inflammatory drugs (NSAIDs), during the trial period for an immune-related disease.
3. Participant has concomitant comorbid skin condition that, in the opinion of the investigator, would interfere with the trial assessments.
4. Participant has history of active TB infection, regardless of treatment status and has signs or symptoms of active TB or evidence of latent tuberculosis infection (LTBI).
5. Participant has active herpes virus infection, including herpes zoster or herpes simplex 1 and 2 or a history of serious herpetic infection.
6. Participant has a history of chronic or recurrent bacterial disease.
7. Participant has a history of opportunistic infections (for example, Pneumocystis jirovecii pneumonia, histoplasmosis, coccidiomycosis).
8. Participant has any clinically significant medical condition, evidence of an unstable clinical condition or vital signs/physical examination/laboratory/ECG abnormality that would, in the opinion of the investigator, put the participant at undue risk or interfere with interpretation of trial results.
9. Participant has any previous exposure to zasocitinib (also known as TAK-279 or NDI-034858) or other TYK2 inhibitors or participated in any trial that included a tyrosine kinase 2 (TYK2) inhibitor, unless participant has documentation of posttrial unblinding that confirms the participant did not receive a TYK2 inhibitor.
10. Participant is not up to date on all required vaccinations according to current immunization guidelines as noted by country-specific pediatric authorities.

Other protocol-defined inclusion/exclusion criteria apply.