A Study of Mezagitamab in Adults With Kidney Condition Called IgA Nephropathy

* 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 (≥) 0.8 gram per gram (g/g) or urine protein excretion (UPE) ≥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 (\>)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 ≥ 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 (≤)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 \>0.5 g/g or UPE \>0.5 g/d calculated from a 24-hour urine collection during the screening period (or pre-screening, if applicable) and eGFR \>30 mL/min/1.73m\^2 at screening based on the CKD-EPI formula. 2. UPCR \<0.8 g/g and UPE ≥0.75 and \<1.0 g/day, by 24-hour urine collection during the screening period (or pre-screening, if applicable) and eGFR \> 30 mL/min/1.73m\^2 at screening based on the CKD-EPI formula. 3. UPCR ≥ 0.8 g/g or UPE ≥ 1.0 g/d by 24-hour urine collection during the screening period (or pre-screening, if applicable) and eGFR ≥25 and ≤30 mL/min/1.73m\^2 at screening based on the CKD-EPI formula. * Exclusion Criteria: A participant who meets any of the following criteria will be excluded from participation in this trial: 1. Kidney biopsy exhibiting significant concomitant renal disease other than IgAN (for example, diabetic nephropathy, lupus nephritis, minimal change disease). 2. Secondary IgAN (such as with significant liver disease, inflammatory bowel disease, and seronegative spondyloarthropathies), and immunoglobulin A (IgA) vasculitis. 3. Evidence of rapidly progressive glomerulonephritis (loss of ≥50% of eGFR within 3 months before the signing of the ICF). 4. Diagnosis of nephrotic syndrome defined as 24-hour proteinuria \>3.5 g/day and hypoalbuminemia (\<3.0 grams per deciliter \[g/dL\]) with or without peripheral edema. 5. Renal or other organ transplantation prior to or expected during the clinical trial. 6. Treatment with oral immunosuppressive agents (including cyclophosphamide, mycophenolate mofetil, cyclosporine, azathioprine, calcineurin inhibitors) or biologic therapy for immunomodulation (including immunomodulatory monoclonal or polyclonal antibodies) within 6 months (both B-cell and non-B-cell directed agents) before signing of the ICF. 7. If the participant has received anti-CD20 treatment, the participant is excluded if either of the following apply: 1. The last dose was received within 6 months before the signing of the ICF. 2. The last dose was received between 6 and 12 months before the signing of the ICF and the participant has a CD19+ count below the lower limit of normal. Note: Participants who have received the last dose of anti-CD20 treatment \>12 months before the signing of the ICF are not excluded from clinical trial participation based on this criterion and are not required to undergo CD19+ testing. 8. Within 4 months of the screening visit, use of either a) systemic corticosteroids at an average dose of 40 milligrams (mg) prednisone equivalent or higher for more than 14 days or b) oral budesonide delayed release capsules. 9. The participant has received a live or live-attenuated vaccine within 4 weeks before signing the ICF or has any live or live-attenuated vaccine planned during the clinical trial. 10. Participation in any other investigational drug trial (including vaccine trial) with receipt of at least 1 dose of investigational drug, or has been exposed to another investigational agent within 4 weeks or 5 half-lives, whichever is longer, before Visit 1. 11. The participant has active infection with hepatitis