The main aim of this study is to evaluate the PK, safety, tolerability and immunogenicity of subcutaneous (SC) administration of TAK-881 in adult and pediatric participants with PIDD and compare them to HYQVIA in participants 16 years old and older.

The participants will be treated with TAK-881/HYQVIA or HYQVIA/TAK-881 with the same dose and dosing interval of immunoglobulin for up to 51 weeks (for participants greater than or equal to [>=]16 years) and only with TAK-881 for up to 27 weeks (for participants aged 2 to less than [<]16 years) as they were treated with another immunoglobulin before enrollment. Participants will need to visit the clinic every 3 or 4 weeks during the duration of the study.

The study consists of a screening epoch, a ramp-up epoch (if needed) and treatment epochs. Participants who have been receiving conventional subcutaneous intravenous immunoglobin G (cIGSC) or intravenous immunoglobulin G (IGIV) before the study, will enter a ramp-up epoch which will start 1, 2, 3 or 4 weeks after the last cIGSC or IGIV pre study dose before screening. Participants who have already been receiving HYQVIA treatment before the study, will directly enter the treatment epochs after screening. Participants aged greater than or equal to [>=]16 years will be randomized at a 1:1 ratio to one of the following treatment sequences: either TAK-881 followed by HYQVIA or HYQVIA followed by TAK-881. Each participant aged >=16 years will complete both crossover epochs. Pediatric participants aged 2 to less than [<]16 years will complete a single arm treatment with the study drug (TAK-881 only).

Inclusion Criteria

• Participant must have a documented diagnosis of a form of primary humoral immunodeficiency involving a defect in antibody formation and requiring IgG replacement, as defined according to the International Union of Immunological Societies (IUIS) Committee.

• Participant is 2 years to <16 years at the time of signing the informed consent form (ICF) for the single arm treatment part of the study OR 16 years or older at the time of signing the ICF for the crossover part of the study.

• Participant has received a stable dose of regular treatment with any IGIV OR HYQVIA with a treatment interval of every 21 or 28 days OR any cIGSC with a treatment interval of every 7 or 14 days over a period of at least 12 weeks prior to screening at a minimum prestudy IgG dose equivalent to 0.3 grams per kilograms per body weight per 4 weeks (g/kg BW/4 weeks) and a maximum dose equivalent to 1 g/kg BW/4 weeks. Over that period, the participant should have been on the same product of IGIV, HYQVIA, or cIGSC. A stable dose is defined as one that deviates less than +-25 percentage (%) from the mean dose for all IgG infusions within this 12-week period prior to screening. Variations in the treatment interval of up to +-5 days for participant with a 28-day treatment interval and of up to +-3 days for participant with a 7, 14, or 21-day treatment interval are acceptable up to the first IP infusion.

• Participant has a serum trough level of IgG greater than (>) 5 grams per liter (g/L) at the following time points:

  1. At screening (sample taken prior to prestudy IgG infusion after signing the ICF) and
  2. Within 12 weeks prior to screening.

• If female of childbearing potential, participant presents with a negative pregnancy test and agrees to employ highly effective form of contraception for the duration of the study.

• Participant, or in the case of minors, legally designated representative(s) is/are willing and able to comply with the requirements of the protocol, including PK blood sampling, for the duration of the study.

Informed Consent

• The participant or, in the case of minors, legally designated representative(s) is/are willing and able to understand and fully comply with study procedures and requirements, in the opinion of the investigator.
• The participant or, in the case of minors, legally designated representative(s) has/have provided informed consent/assent, if applicable, (that is, in writing, documented via a signed and dated ICF and/or eConsent, if available), and any required privacy authorization prior to the initiation of any study procedures.

Exclusion Criteria

• Participant has a known history of a positive result or is positive at screening for one or more of the following: hepatitis B surface antigen (HBsAg), polymerase chain reaction (PCR) for hepatitis C virus (HCV), PCR for human immunodeficiency virus (HIV) Type 1/2. Cured participants with a history of hepatitis C infection who have a negative PCR test at screening is eligible.

• Abnormal laboratory values at screening meeting any one of the following criteria (abnormal tests may be repeated once to determine if they are persistent):

  1. Persistent alanine aminotransferase (ALT) and aspartate aminotransferase (AST) >2.5* the upper limit of normal (ULN) for the testing laboratory.
  2. Persistent severe neutropenia (defined as an absolute neutrophil count [ANC] less than and equal to (<=) 500 per cubic millimeter (/mm^3).

• Known history of chronic kidney disease or estimated glomerular filtration rate (eGFR) of <60 milliliter per minute per 1.73 square meter (mL/min/1.73m^2) at screening.

• Participant has anemia that would preclude phlebotomy for laboratory studies, according to standard practice at the site, at the discretion of the investigator.

• Participant has a known history of hypersensitivity or persistent reactions (urticaria, breathing difficulty, severe hypotension, or anaphylaxis) following IV immunoglobulin, SC immunoglobulin, and/or immune serum globulin infusions.

• Participants with a known systemic hypersensitivity to any of the excipients of TAK-881/HYQVIA in accordance with the Investigator's Brochure (IB)/package insert/Summary of Product Characteristics (SmPC).

• Known substance or prescription drug abuse within 12 months of screening.

• Participant has immunoglobulin A (IgA) deficiency (IgA less than 0.07 g/L) associated with known anti-IgA antibodies and a history of hypersensitivity.

• Participant has a known systemic hypersensitivity to hyaluronidase or rHuPH20.

• Participant has active infection and is receiving antibiotic therapy for the treatment of infection at the time of screening.

• Participant has a bleeding disorder, or a platelet count less than 20,000 per microliter (mcL), or in the opinion of the investigator, would be at significant risk of increased bleeding or bruising as a result of immune globulin subcutaneous (IGSC) therapy.

• Treatment with immunosuppressants including chemotherapeutic agents, immunomodulators, and long-term systemic corticosteroid (defined as a daily dose of >1 mg of prednisone equivalent/kg/day for >30 days) within 12 weeks prior to screening. Short or intermittent courses (<=10 days) of corticosteroids are allowed.

• Live-attenuated viral vaccination within 12 weeks prior to screening.

• History or current diagnosis of thrombotic episodes; venous thrombus that occurred in association with a medical device >2 years prior to screening are allowed.

• Participant has severe dermatitis that would preclude adequate sites for safe product administration in the opinion of the investigator.

• Participant has a medical condition, laboratory finding, or physical examination finding that precludes participation, or with clinical evidence of any significant acute or chronic disease that, in the opinion of the investigator, may interfere with successful completion of the study or place the participant at undue medical risk.

• Participant has participated in another clinical study involving an investigational product (IP) or investigational device within 30 days prior to screening.

• Participant is scheduled to participate in another clinical study involving an IP (except for participants scheduled to enroll in a long-term follow-up study with TAK-881) or investigational device during the course of this study.

• Participant is a family member or employee of the investigator or the investigator's site staff.

• If female, participant is pregnant or lactating at the time of screening.

Takeda provides access to the de-identified individual participant data (IPD) for eligible studies to aid qualified researchers in addressing legitimate scientific objectives (Takeda's data sharing commitment is available on https://clinicaltrials.takeda.com/takedas-commitment?commitment=5). These IPDs will be provided in a secure research environment following approval of a data sharing request, and under the terms of a data sharing agreement.

IPD from eligible studies will be shared with qualified researchers according to the criteria and process described on https://vivli.org/ourmember/takeda/ For approved requests, the researchers will be provided access to anonymized data (to respect patient privacy in line with applicable laws and regulations) and with information necessary to address the research objectives under the terms of a data sharing agreement.