Calculating a Moving Average MySQL?

Starting from MySQL 8, you should use window functions for this. Using the window RANGE clause, you can create a logical window over an interval, which is very powerful. Something like this:

SELECT
  date,
  close,
  AVG (close) OVER (ORDER BY date DESC RANGE INTERVAL 9 DAY PRECEDING)
FROM tbl
WHERE date <= DATE '2002-07-05'
AND name_id = 2
ORDER BY date DESC

For example:

WITH t (date, `close`) AS (
  SELECT DATE '2020-01-01', 50 UNION ALL
  SELECT DATE '2020-01-03', 54 UNION ALL
  SELECT DATE '2020-01-05', 51 UNION ALL
  SELECT DATE '2020-01-12', 49 UNION ALL
  SELECT DATE '2020-01-13', 59 UNION ALL
  SELECT DATE '2020-01-15', 30 UNION ALL
  SELECT DATE '2020-01-17', 35 UNION ALL
  SELECT DATE '2020-01-18', 39 UNION ALL
  SELECT DATE '2020-01-19', 47 UNION ALL
  SELECT DATE '2020-01-26', 50
)
SELECT
  date,
  `close`,
  COUNT(*) OVER w AS c,
  SUM(`close`) OVER w AS s,
  AVG(`close`) OVER w AS a
FROM t
WINDOW w AS (ORDER BY date DESC RANGE INTERVAL 9 DAY PRECEDING)
ORDER BY date DESC

Leading to:

date      |close|c|s  |a      |
----------|-----|-|---|-------|
2020-01-26|   50|1| 50|50.0000|
2020-01-19|   47|2| 97|48.5000|
2020-01-18|   39|3|136|45.3333|
2020-01-17|   35|4|171|42.7500|
2020-01-15|   30|4|151|37.7500|
2020-01-13|   59|5|210|42.0000|
2020-01-12|   49|6|259|43.1667|
2020-01-05|   51|3|159|53.0000|
2020-01-03|   54|3|154|51.3333|
2020-01-01|   50|3|155|51.6667|

If you want the moving average for each date, then try this:

SELECT date, SUM(close),
       (select avg(close) from tbl t2 where t2.name_id = t.name_id and datediff(t2.date, t.date) <= 9
       ) as mvgAvg
FROM tbl t
WHERE date <= '2002-07-05' and
      name_id = 2
GROUP BY date
ORDER BY date DESC

It uses a correlated subquery to calculate the average of 9 values.