Check that the Enable pins on the L293 (1 and 9) are properly connected to 5V.
Since no battery is shown I assume it's powered by USB? It's a really bad idea running motors from the Arduino 5V pin, it can't deliver enough current. Take out the link to L293 pin8 and try it with a separate battery maybe 4 x AA cells +ve connected to pin 8 and -ve to Arduino ground.
And just to be certain...you're saying that either motor will run provided that it's connected to pins 5 and 6? The motor connected to 9 and 10 never works?
Steve