SELECT 
  a.category_id 
FROM 
  cscart_categories as a 
  LEFT JOIN cscart_categories as b ON b.category_id IN (669) 
WHERE 
  a.id_path LIKE CONCAT(b.id_path, '/%')

Query time 0.00077

JSON explain

{
  "query_block": {
    "select_id": 1,
    "table": {
      "table_name": "b",
      "access_type": "const",
      "possible_keys": ["PRIMARY", "p_category_id"],
      "key": "PRIMARY",
      "key_length": "3",
      "used_key_parts": ["category_id"],
      "ref": ["const"],
      "rows": 1,
      "filtered": 100
    },
    "table": {
      "table_name": "a",
      "access_type": "ALL",
      "rows": 654,
      "filtered": 100,
      "attached_condition": "a.id_path like <cache>(concat('669','/%'))"
    }
  }
}