Inclusion Criteria:

To be eligible to participate in this trial, participants must meet all the following criteria:

1. Either UPCR greater than or equal to (greater than or equal to) 0.8 gram per gram (g/g) or urine protein excretion (UPE) greater than or equal to 1 grams per day (g/day), calculated from at least one 24-hour urine collection during the screening period (or pre-screening, if applicable) (only applicable for the main trial).
2. eGFR greater than (greater than) 30 milliliters per minute per 1.73 meter square (mL/min/1.73m^2) at screening based on the Chronic Kidney Disease Epidemiology Collaboration (CKD-EPI) formula (only applicable for the main trial).
3. No prior exposure to anti- cluster of differentiation 38 (CD38) therapy period (except for open-label cohort participants meeting Inclusion Criterion No. 10.a).
4. The participant is aged greater than or equal to 18 years or the local legal age as applicable.
5. The participant (and the participant's legally acceptable representative, as applicable per local regulations or determination) has provided informed consent (that is, in writing, documented via a signed and dated informed consent form [ICF]) and any required privacy authorization before the initiation of any clinical trial procedures.
6. Diagnosis of primary immunoglobulin A nephropathy (IgAN) supported by a renal biopsy report that is dated more recently than 10 years before the signing of the informed consent for the clinical trial. The redacted report must be made available for review. A renal biopsy must be performed during screening for participants without a biopsy report within 10 years.
7. Participants must be on stable renin-angiotensin-aldosterone system (RAAS) inhibitor therapy with an angiotensin-converting enzyme inhibitor (ACE-I) and/or angiotensin receptor blocker (ARB) or endothelin receptor antagonist (ERA) or mineralocorticoid receptor antagonist (MRA) agent for at least 12 weeks before signing the ICF with dosing at the maximally tolerated or labeled dose as determined by the investigator, with the intent to continue stable dosing during the clinical trial. Those intolerant of RAAS inhibitor therapy are potentially eligible after consultation with the medical monitor. Intolerance is defined as a documented side effect causing discontinuation of the therapy.
8. Resting blood pressure less than or equal to (less than or equal to)150 millimeters of mercury (mmHg) systolic and ≤100 mmHg diastolic.
9. Female participants of childbearing potential who are not pregnant during screening (confirmed by negative serum human chorionic gonadotropin [hCG]) and on Visit 1 before first dose of trial intervention (confirmed by negative urine pregnancy test).
10. Any one of the following (only applicable for participants in the open-label cohort):

1. Participants in Trial TAK-079-1006 who completed the Week 96 visit or the retreatment period with either UPCR greater than 0.5 g/g or UPE greater than or equal to 0.5 g/d calculated from a 24-hour urine collection during the screening period (or pre-screening, if applicable) and eGFR greater than 30 mL/min/1.73m^2 at screening based on the CKD-EPI formula.
2. UPCR less than 0.8 g/g and UPE greater than or equal to 0.75 and less than 1.0 g/day, by 24-hour urine collection during the screening period (or pre-screening, if applicable) and eGFR greater than 30 mL/min/1.73m^2 at screening based on the CKD-EPI formula.
3. UPCR greater than or equal to 0.8 g/g or UPE greater than or equal to 1.0 g/d by 24-hour urine collection during the screening period (or pre-screening, if applicable) and eGFR greater than or equal to 25 and less than or equal to 30 mL/min/1.73m^2 at screening based on the CKD-EPI formula.

Note: The complete list of exclusion criteria can be found in the protocol.