The main aim of this study is to learn if TAK-881 is safe and well tolerated during long-term use in adults and children with PIDD who are eligible and completed study TAK-881-3001 (NCT05755035).

The participants will receive the first dose of TAK-881 immunoglobulin (IgG) infusion at the same dosage as the last dose administered in study TAK-881-3001 (NCT05755035). After the first TAK-881 infusion in study TAK-881-3002, the dosing interval (can be adjusted by participant/caregiver) and/or the dose of TAK-881 can be adjusted by the study doctor to every 2, 3 or 4 weeks at scheduled site visits.

Participants will visit the clinic every 12 weeks until the End of Study (EOS) visit.

The study consists of two Epochs: Epoch 1 and Epoch 2. In Epoch 1 eligible participants will have the opportunity to receive TAK-881 for a minimum of 6 months. After 6 months in study TAK-881-3002, the results of the anti-rHuPH20 binding antibody assay from either study TAK-881-3001 (NCT05755035) or study TAK-881-3002 (Epoch 1) will be used to determine the next steps in the study:

• Epoch 1: Participants with an anti-rHuPH20 antibody titer less than [<] 1:160 will complete the EOS visit on the day of the first non-investigational product (IP) infusion following Study Visit Week 25.
• Epoch 2: Participants with an anti-rHuPH20 antibody titer greater than or equal to [>=] 1:160 at any time point during Study TAK-881-3001 (NCT05755035) and/or study TAK-881-3002 (Epoch 1), will continue with an additional two years of TAK-881 treatment and undergo safety follow-up and will complete the EOS visit on the day of the first non-IP infusion following study Visit Week 121.

Inclusion Criteria

Participants who meet ALL of the following criteria are eligible for this study:

• Participant must have completed Study TAK-881-3001 (NCT05755035).
• Participant or, in the case of minors, legally designated representative(s) is/are willing and able to comply with the requirements of the protocol.
• Participant or, in the case of minors, legally designated representative(s) has/have provided informed consent/assent (if applicable), including providing consent for use of the investigational medical device (in US sites only), prior to the initiation of any study procedures.

Exclusion Criteria

• Participant has a serious medical condition such that the participant's safety or medical care would be impacted by participation in this long-term follow-up study.

• New medical condition that developed during participation in Study TAK-881-3001 (NCT05755035), that, in the judgment of the investigator, could increase risk to the participant or interfere with the evaluation of TAK-881 and/or conduct of the study.

• Participant was enrolled in another clinical study involving an investigational product (IP) or investigational device within 30 days prior to screening (with the exception of Study TAK-881-3001 [NCT05755035]).

• Participant is scheduled to participate in another clinical study involving an IP 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.

• Participants with potential to become pregnant who meet any one of the following criteria:-

  1. Participant has a positive pregnancy test.
  2. Participant does not agree to employ a highly effective form of contraception for the duration of the study.

• If participant is pregnant or lactating at the time of screening or intends to become pregnant or begin lactating during the study.

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.