It seems that @my_A contains a string which is already JSON-encoded. Passing it to encode_json double-encodes it, ie escapes all quotes with backslashes.

If that is the case, you can produce your desired result through a simple text operation, like

my $json_output = '{"JSON1:[" . join(',', @my_A) .']}';
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of you have to decode the existing JSON string before inserting it into a data structure that you pass to encode_json.