I believe Hard Times (Bellflower) has a handicapped Thursday night weekly tournament, and has monthly (open) 9-ball and one pocket tournaments, first sat/sun of the month.
What part of LA are you located in?
I'm sure there are other weekly tournaments around Los Angeles, there's probably something at Metro Billiards in Carson or Lakewood Family Billiards. If you don't mind driving a little, Cue Ball in Alhambra has a Sunday night handicapped 9-ball tournament. You've also got House of Billiards in Santa Monica, House of Billiards Sherman Oaks, etc.