For historical child maintenance orders from 1929 to 1939 in Manchester, you could try the following:
Manchester Archives and Local Studies: They may hold court records or other relevant documents from that period. Contact them directly to inquire about their holdings.
National Archives (UK): They might have records of magistrates' courts or other legal documents related to maintenance orders.
Local Magistrates' Court Records: Maintenance orders were often handled by magistrates' courts. Check if records from the Manchester courts of that era are archived locally or transferred to a central repository.
Electoral Rolls or Census Records: These might help trace the mother and child’s residence, which could narrow down the court jurisdiction.
Newspapers: Local newspapers from that time might have reported on court cases, including maintenance orders.
Since you have limited details, it’s worth reaching out to these institutions with as much information as possible.