Problem Importing JSON data in CosmosDB from Azure Data Factory Copy

Per my knowledge, no such features could help you convert string data to object format in adf cosmos db configuration.

So, as workaround, I suggest you using Azure Function Cosmos DB Trigger to process every document when they imported into database. Please refer to my function code:

using System.Collections.Generic;
using Microsoft.Azure.Documents;
using Microsoft.Azure.WebJobs;
using Microsoft.Azure.WebJobs.Host;
using Newtonsoft.Json.Linq;
using System;
using Microsoft.Azure.Documents.Client;

namespace TestADF
{
    public static class Function1
    {
        [FunctionName("Function1")]
        public static void Run([CosmosDBTrigger(
            databaseName: "db",
            collectionName: "item",
            ConnectionStringSetting = "documentdbstring",
            LeaseCollectionName = "leases")]IReadOnlyList<Document> input, TraceWriter log)
        {
            if (input != null && input.Count > 0)
            {
                log.Verbose("Start.........");
                String endpointUrl = "https://***.documents.azure.com:443/";
                String authorizationKey = "key";
                String databaseId = "db";
                String collectionId = "item";

                DocumentClient client = new DocumentClient(new Uri(endpointUrl), authorizationKey);

                for (int i = 0; i < input.Count; i++)
                {
                    Document doc = input[i];
                    if ((doc.GetPropertyValue<String>("Value") == null) || (!doc.GetPropertyValue<String>("Value")))
                    {                       
                        String V= doc.GetPropertyValue<String>("Value");
                        JObject obj = JObject.Parse(V);

                        doc.SetPropertyValue("Value", obj );

                        client.ReplaceDocumentAsync(UriFactory.CreateDocumentUri(databaseId, collectionId, doc.Id), doc);

                        log.Verbose("Update document Id " + doc.Id);
                    }

                }
            }
        }
    }
}