SELECT 
  a.category_id 
FROM 
  cscart_categories as a 
  LEFT JOIN cscart_categories as b ON b.category_id IN (577) 
WHERE 
  a.id_path LIKE CONCAT(b.id_path, '/%')

Query time 0.00166

JSON explain

{
  "query_block": {
    "select_id": 1,
    "cost_info": {
      "query_cost": "37.54"
    },
    "nested_loop": [
      {
        "table": {
          "table_name": "b",
          "access_type": "const",
          "possible_keys": [
            "PRIMARY",
            "p_category_id"
          ],
          "key": "PRIMARY",
          "used_key_parts": [
            "category_id"
          ],
          "key_length": "3",
          "ref": [
            "const"
          ],
          "rows_examined_per_scan": 1,
          "rows_produced_per_join": 1,
          "filtered": "100.00",
          "cost_info": {
            "read_cost": "0.00",
            "eval_cost": "0.10",
            "prefix_cost": "0.00",
            "data_read_per_join": "3K"
          },
          "used_columns": [
            "category_id",
            "id_path"
          ]
        }
      },
      {
        "table": {
          "table_name": "a",
          "access_type": "ALL",
          "rows_examined_per_scan": 347,
          "rows_produced_per_join": 38,
          "filtered": "11.11",
          "cost_info": {
            "read_cost": "2.84",
            "eval_cost": "3.86",
            "prefix_cost": "37.54",
            "data_read_per_join": "120K"
          },
          "used_columns": [
            "category_id",
            "id_path"
          ],
          "attached_condition": "(`softwarepirmam_hewadelivard_cscart_4`.`a`.`id_path` like <cache>(concat('577','/%')))"
        }
      }
    ]
  }
}