There is a good choice of eating places in downtown LA.
The El Patio is a very Spanish with the biggest BBQ i have ever seen. The food is GR8 off the bbq with a large portion of chicken at 4.50 euro. The house vino from Jumilla which has just won a silver medal is 5* at only 4.20 euro
In the precint next to the Mercadona supermarket you have La Diva and Bayside both more international. Good food but far more expensive than El Patio.
Down by the arch near the Oasis you have the Roblemar, which has the best selection of fresh fish/shell fish you will find any where in Spain. This is also a lovely Tapas bar next door with fantastic selection of Serrano hams and all kinds of fishy things.
In the centre of LA is La Encarnation which is also a boutique hotel. This is a class place with GR8 atmosphere, both in summer and autumn. In fact in July we joined in on one of their dinner dances on the outside terrace. Live music and a real knees up Spanish style.