Do you need the genre names? If you can do with just the genre_id, you can eliminate the second join. (You can fill in the genre name later, in the UI, using a cache).
What indexes do you have?
You probably want
create index idx_movies on movies
(mov_added, mov_type, mov_status, mov_incomplete)
and most certainly the join index
create index ind_genres_movies on _genres
(gen_mov_id, gen_cat_id)
Can you post the output of EXPLAIN? i.e. put EXPLAIN in front of the SELECT and post the results.
I've had quite a few wins with using SELECT STRAIGHT JOIN and ordering the tables according to their size.
STRAIGHT JOIN stops mysql guess which order to join tables and does it in the order specified so if you use your smallest table first you can reduce the amount of rows being joined.
I'm assuming you have indexes on mov_id, gen_movieid, gen_catid and grenre_id?
The 'using temporary, using filesort' is from the group concat distinct genres.genre_name.
Trying to get a distinct on a column without an index will cause a temp table to be used. Try adding an index on genre_name column.