As far as I know there is not a directed way to do it but you can to use another way: Using aggregation stages into update query:
The trick is to use $slice. Check how it works here.
This example is concating two arrays, each array is using a different syntax, note that the documentation says:
$slice
Returns a subset of an array.
$slice has one of two syntax forms:
The following syntax returns elements from either the start or end of the array:
{ $slice: [ <array>, <n> ] }
The following syntax returns elements from the specified position in the array:
{ $slice: [ <array>, <position>, <n> ] }
So, in this example:
- The first $sliceis getting the elements from the start to the index.
- The second $sliceis getting the elements from the index to the end.
If you concatenate these two arrays you get the same array without the index.
db.collection.update({
  "_id": 1
},
[
  {
    "$set": {
      "array": {
        "$concatArrays": [
          {
            "$slice": [
              "$array",
              YOUR_INDEX
            ]
          },
          {
            "$slice": [
              "$array",
              {
                "$add": [
                  1,
                  YOUR_INDEX
                ]
              },
              {
                "$size": "$array"
              }
            ]
          }
        ]
      }
    }
  }
])
Example here