edit: As stated in the comments - DISTINCT
is not required because the auctions table can only be associated with (at most) one user table row and one row in the inner query.
You may want to try this. The outer query's GROUP BY is replaced with DISTINCT since you don't have any aggregate function. The inner query, was replaced by a query to find the smallest media_id per auction_id, then JOINed back to get the media_url. (Since I didn't know if the media_id and auction_id were a composite unique key, I used the same WHERE clause to help eliminate potential duplicates.)
SELECT
a.auction_id,
a.name,
auction_media.media_url
FROM auctions AS a
LEFT JOIN users AS u
ON u.user_id=a.owner_id
INNER JOIN (SELECT auction_id, MIN(media_id) AS media_id
FROM auction_media
WHERE media_type=1
AND upload_in_progress=0
GROUP BY auction_id) AS media
ON a.auction_id=media.auction_id
INNER JOIN auction_media
ON auction_media.media_id = media.media_id
AND auction_media.auction_id = media.auction_id
AND auction_media.media_type=1
AND auction_media.upload_in_progress=0
WHERE a.hpfeat=1
AND a.active=1
AND a.approved=1
AND a.closed=0
AND a.creation_in_progress=0
AND a.deleted=0
AND (a.list_in='auction' OR u.shop_active='1');