How to add a field to a JSON object with the jq command?

This type of update is where the magic of += comes into play. With your input, the following invocation:

jq '.module.data.item_i77f664a2.fields += {"operation":"delete"}'

produces the output you indicate you want:

{
  "success": true,
  "module": {
    "data": {
      "item_i77f664a2": {
        "id": "i77f664a2",
        "tag": "item",
        "fields": {
          "cartItemId": 2012636322,
          "operation": "delete"
        },
        "type": "biz"
      }
    }
  }
}

However, I am not sure that this will produce what you want in similar circumstances, as you have referenced “item_i77f61ee2”.