Complex sentence: U.S. and British authorities believed Anwar al-Awlaki of AQAP was behind the bombing attempts, and that the bombs were most likely constructed by AQAP's main explosives expert, Ibrahim Hassan al-Asiri.
Split into simpler sentences: U.S. and British authorities had believed that AQAP, and specifically Anwar al-Awlaki, were behind the bombing attempts. They also believed the bombs were most likely constructed by AQAP's main explosives expert, Ibrahim Hassan al-Asiri.