A Study of TAK-279 in Participants With Moderate-to-Severe Plaque Psoriasis

Main Inclusion Criteria: Part A: * Participant is willing and able to understand and fully comply with study procedures and requirements (including digital tools and applications), in the opinion of the investigator. * Participant has provided written informed consent and any required privacy authorization before the initiation of any study procedures. * Participant is aged 18 years or older at the time of consent. * Participant has a diagnosis of chronic plaque psoriasis for \>=6 months prior to the screening visit. * 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. * Participant has moderate-to-severe plaque psoriasis as defined by a PASI score \>=12 and a sPGA score \>=3 at screening and Day 1. * Participant has plaque psoriasis covering \>=10% of his or her total BSA at screening and Day 1. * Participant must be a candidate for phototherapy or systemic therapy. Part B: \- Participant has completed 52 weeks of treatment (TAK-279-3001 or Part A) or 60 weeks of treatment (TAK-279-3002), or 16 weeks of treatment (TAK-279-PsO-3004) in their parent study. Main Exclusion Criteria Part A: * Participant has evidence of non-plaque 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. * Participant requires systemic treatment, other than nonsteroidal anti-inflammatory drugs, during the trial period for an immune-related disease (e.g., inflammatory bowel disease). * Participant has any clinically significant medical condition, evidence of an unstable clinical condition (e.g., cardiovascular, renal, hepatic, hematologic, gastrointestinal, endocrine, pulmonary, or immunologic), or vital signs/physical/laboratory/ECG abnormality that would, in the opinion of the investigator, put the participant at undue risk or interfere with interpretation of study results. These include but are not limited to: 1. Participant has a history of known or suspected condition/illness that is consistent with compromised immunity, including but not limited to any identified congenital or acquired immunodeficiency; splenectomy. 2. Participant had a major surgery within 60 days prior to Day 1 or has a major surgery planned during the study. 3. Participant has unstable, poorly controlled, or severe hypertension at screening, confirmed by 2 repeat assessments. 4. Participant has a history of Class III or IV congestive heart failure as defined by New York Heart Association criteria. 5. Participant has a history of cancer or lymphoproliferative disease, with the exception of successfully treated nonmetastatic cutaneous squamous cell or basal cell carcinoma and/or localized carcinoma in situ of the cervix. 6. For participants with asthma, chronic obstructive pulmonary disease, or other pulmonary illnesses, participant has been hospitalized in the past 3 months, has ever required intubation for treatment, currently requires oral corticosteroids, or has required more than 1 course of oral corticosteroids within 6 months prior to Day 1. 7. Participant has any of the following cardiovascular disease history: A new diagnosis of atrial fibrillation or an episode of atrial fibrillation with rapid ventricular response or other dysrhythmia, non-acute cardiac hospitalization (e.g., pacemaker implantation), pulmonary embolism, or deep venous thrombosis within the past 6 months prior to screening. Any history of cerebrovascular event, myocardial infarction, coronary stenting, or aortocoronary bypass surgery. If, however, the investigator determines there are no suitable treatment alternatives available for the participant and it has been at least 6 months since the occurrence of any such event, the participant may enroll. 8. Participant has ECG abnormalities that are considered clinically significant and would pose an unacceptable risk to the participant if he or she participated in the study, in the opinion of the investigator. 9. Participant has significant/uncontrolled psychiatric illness, in the opinion of the investigator. 10. Participant has any lifetime history of suicidal ideation, suicidal behavior, or suicidal attempts by 1) medical history; or 2) by Columbia-Suicide Severity Rating Scale (C-SSRS) documentation at screening or by answering "yes" to Question 5 for suicidal ideation on the C-SSRS at screening; or 3) is clinically deemed to have a suicide risk by the investigator. 11. Participant has a patient health questionnaire - 8 (PHQ-8) score of 15 or above at screening. 12. Participant has a history of clinically significant drug or alcohol abuse within 12 months prior to Day 1. * Participant has received any of the following biologics or biosimilar versions within the time frame indicated or 5 half-lives, whichever is longer: 1. Antibodies to interleukin (IL) -12/-23, IL-17, or IL-23 (eg, ustekinumab, secukinumab, tildrakizumab, ixekizumab, or guselkumab) within 6 months prior to Day 1. 2. Tumor necrosis factor inhibitor(s) (e.g., etanercept, adalimumab, infliximab, certolizumab) within 2 months prior to Day 1. 3. Agents that modulate integrin pathways to impact lymphocyte trafficking (e.g., natalizumab) or agents that modulate B cells or T cells (e.g., alemtuzumab, abatacept, or visilizumab) within 3 months prior to Day 1. 4. Rituximab or other immune cell-depleting therapy within 6 months prior to Day 1. * Participant has any previous exposure to TAK-279 (also known as NDI-034858) or other TYK2 inhibitors, including deucravacitinib, or participant participated in any study that included a TYK2 inhibitor (e.g., deucravacitinib, VTX958, GLPG3667, etc.), unless the participant has documentation of post-trial unblinding that confirms the partcipant did not receive a TYK2 inhibitor. * Participant has a known or suspected allergy to TAK-279 or any of its components. Part B: * Participant has completed the parent study or Part A but was permanently discontinued from treatment. * Participant had evidence of significant noncompliance with study visits or study drug in the parent study or Part A, as defined in the parent study protocol or in the opinion of the investigator. * Participant has met criteria for termination from the parent study or Part A, regardless of whether or not the participant was terminated from the parent study or Part A. * Participant has developed evidence of non-plaque psoriasis (erythrodermic, pustular, predominantly guttate psoriasis, predominantly inverse, or drug-induced psoriasis) since enrollment in the parent study or Part A. * Participant has developed a concomitant comorbid skin condition that, in the opinion of the investigator, would interfere with the study assessments. * Participant has received a prohibited psoriasis treatment during the parent study or Part A, whether or not that treatment was documented as a concomitant medication and is expected to continue that treatment.