How to inject additional parameters to AWS Lambda Function when it invoked by Step Function

I have defined AWS Step Function using Terraform and that Step Function has 2 Tasks. And if any error occurs in the 1st task, it goes to the CATCH block, and from there we are calling the 2nd Task. The 2nd Task triggers a Java Lambda Handler that handles the error that occurred in the 1st task.

I have already added the error I got in the 1st task to the State using "ResultPath" as shown in the below code and the Java Lambda Handler in the 2nd Task is reading that error and processing it. The entire state which contains the error will be passed to the "Java Lambda Error Handler" as a Parameter as shown in the below code.

So my question here is: Once the control reaches the "Second Task", I want to add a couple of additional fields that should go as inputs to the Lambda Handler. I want to add a couple of additional fields to the existing State (that contains error details) that are already going as input to the Lambda Handler.

Could someone please help me with how I can add a couple of additional fields in my below code?

"States": {
    "First_Task": {
        ...
        ...
        "Catch": [
         {
            "ErrorEquals": [
               "States.All"
            ], 
            "Next": "Second_Task",
            "ResultPath": "$.error"
         }
        ],
        "OutputPath": "$.Payload", 
        "End": true
    },
    "Second_Task": {
       "Type": "Task",
       "Resource": "arn:aws:states:::lambda:invoke",
       "OutputPath": "$.Payload", 
       "Parameters": {
          "Payload.$": "$",  // -> here we are passing the error details to Lambda
          // -> Can I pass some additional fields here as input to the Lambda Handler
          "FunctionName": "${<ErrorHandlerLambdaFunctionName>}"
       },
       ......
    }
}

You can use the Parameters fields to construct the payload that you send to your Lambda function by composing from the input, Step Functions Context Object, and literal values. See an example state machine below:

{
  "StartAt": "FIrst Task",
  "States": {
    "FIrst Task": {
      "Type": "Task",
      "Resource": "arn:aws:states:::lambda:invoke",
      "OutputPath": "$.Payload",
      "Parameters": {
        "Payload.$": "$",
        "FunctionName": "arn:aws:lambda:ca-central-1:123456789012:function:function_that_will_fail"
      },
      "End": true,
      "Catch": [
        {
          "ErrorEquals": [
            "States.ALL"
          ],
          "Next": "Second Task",
          "ResultPath": "$.error"
        }
      ]
    },
    "Second Task": {
      "Type": "Task",
      "Resource": "arn:aws:states:::lambda:invoke",
      "OutputPath": "$.Payload",
      "Parameters": {
        "Payload": {
          "error.$": "$.error",
          "full_input.$": "$",
          "step_functions_execution_id.$": "$$.Execution.Id",
          "static_stuff": {
            "some_array": [
              1,
              2,
              3,
              4
            ],
            "some_key": "some_value"
          }
        },
        "FunctionName": "arn:aws:lambda:ca-central-1:123456789012:function:return_what_was_passed"
      },
      "End": true
    }
  }
}

When I execute this with a simple comment as input, this is what I'll get back

{
  "static_stuff": {
    "some_array": [
      1,
      2,
      3,
      4
    ],
    "some_key": "some_value"
  },
  "step_functions_execution_id": "arn:aws:states:ca-central-1:123456789012:execution:so-inject-data:faf01f02-10f0-4207-94f0-2b5b1c2b5ba2",
  "full_input": {
    "Comment": "This is a comment for input to the execution",
    "error": {
      "Error": "ValueError",
      "Cause": "{\"errorMessage\":\"This function failed\",\"errorType\":\"ValueError\",\"requestId\":\"1e569a61-38e6-432f-8f77-bae1331d6d11\",\"stackTrace\":[\"  File \\\"/var/task/lambda_function.py\\\", line 4, in lambda_handler\\n    raise ValueError(\\\"This function failed\\\")\\n\"]}"
    }
  },
  "error": {
    "Error": "ValueError",
    "Cause": "{\"errorMessage\":\"This function failed\",\"errorType\":\"ValueError\",\"requestId\":\"1e569a61-38e6-432f-8f77-bae1331d6d11\",\"stackTrace\":[\"  File \\\"/var/task/lambda_function.py\\\", line 4, in lambda_handler\\n    raise ValueError(\\\"This function failed\\\")\\n\"]}"
  }
}