SELECT 
  p.product_id, 
  ifnull(
    count(dp.post_id), 
    0
  ) as discussion_amount_posts 
FROM 
  cscart_discussion 
  INNER JOIN cscart_products as p ON (
    cscart_discussion.object_id = p.product_id
  ) 
  INNER JOIN cscart_discussion_posts as dp ON (
    cscart_discussion.thread_id = dp.thread_id 
    AND cscart_discussion.object_type = 'P' 
    AND cscart_discussion.company_id = 1
  ) 
WHERE 
  dp.status = 'A' 
  and p.product_id in (
    618, 671, 842, 866, 871, 1051, 1513, 584, 
    674, 1696
  ) 
GROUP BY 
  p.product_id

Query time 0.00119

JSON explain

{
  "query_block": {
    "select_id": 1,
    "cost_info": {
      "query_cost": "29.17"
    },
    "grouping_operation": {
      "using_filesort": false,
      "nested_loop": [
        {
          "table": {
            "table_name": "p",
            "access_type": "range",
            "possible_keys": [
              "PRIMARY"
            ],
            "key": "PRIMARY",
            "used_key_parts": [
              "product_id"
            ],
            "key_length": "3",
            "rows_examined_per_scan": 10,
            "rows_produced_per_join": 10,
            "filtered": "100.00",
            "using_index": true,
            "cost_info": {
              "read_cost": "3.17",
              "eval_cost": "2.00",
              "prefix_cost": "5.17",
              "data_read_per_join": "59K"
            },
            "used_columns": [
              "product_id"
            ],
            "attached_condition": "(`eshopuga_upgr_4122`.`p`.`product_id` in (618,671,842,866,871,1051,1513,584,674,1696))"
          }
        },
        {
          "table": {
            "table_name": "cscart_discussion",
            "access_type": "eq_ref",
            "possible_keys": [
              "PRIMARY",
              "object_id",
              "company_id"
            ],
            "key": "object_id",
            "used_key_parts": [
              "object_id",
              "object_type",
              "company_id"
            ],
            "key_length": "10",
            "ref": [
              "eshopuga_upgr_4122.p.product_id",
              "const",
              "const"
            ],
            "rows_examined_per_scan": 1,
            "rows_produced_per_join": 10,
            "filtered": "100.00",
            "cost_info": {
              "read_cost": "10.00",
              "eval_cost": "2.00",
              "prefix_cost": "17.17",
              "data_read_per_join": "240"
            },
            "used_columns": [
              "thread_id",
              "object_id",
              "object_type",
              "company_id"
            ]
          }
        },
        {
          "table": {
            "table_name": "dp",
            "access_type": "ref",
            "possible_keys": [
              "thread_id",
              "thread_id_2"
            ],
            "key": "thread_id",
            "used_key_parts": [
              "thread_id"
            ],
            "key_length": "3",
            "ref": [
              "eshopuga_upgr_4122.cscart_discussion.thread_id"
            ],
            "rows_examined_per_scan": 1,
            "rows_produced_per_join": 1,
            "filtered": "10.00",
            "cost_info": {
              "read_cost": "10.00",
              "eval_cost": "0.20",
              "prefix_cost": "29.17",
              "data_read_per_join": "448"
            },
            "used_columns": [
              "post_id",
              "thread_id",
              "status"
            ],
            "attached_condition": "(`eshopuga_upgr_4122`.`dp`.`status` = 'A')"
          }
        }
      ]
    }
  }
}