It's my experience that people who bring up Anwar al-Awlaki and his son --- particularly when using the term "US citizen" in the same breath --- rarely know anything specific about what actually happened.
To start with, al-Awlaki and his son Abdulrahman weren't killed together, but at separate times.
Abdulrahman was never targeted. He was killed a month later because he was for some reason in the same cafe as Ibrahim al-Banna, the day-to-day operational leader of AQAP. al-Banna was the target of the attack. The USG had no reason to know of Abdulrahman's presence in Shabwa, Yemen, which wasn't under the control of the Yemeni government and continues to be the site of bloody conflicts between militants and the Yemeni state. What was Abdulrahman doing there? Super-parent Anwar al-Awlaki brought him there, to an AQAP war zone that was hosting conflicts not just with the US but with its own host country.
Anwar al-Awlaki was targeted by the US. al-Awlaki was a US citizen. But, like the tens of Americans who left the US to fight for Hitler in World War 2, al-Awlaki deliberately placed himself outside the protections of his citizenship and from that vantage point tried to wage war on his home country. al-Awlaki built up a serial murderer's resume as a sponsor, planner, and coordinator of multiple attacks on civilians in the US and UK. The only reason anyone questions his role in those attacks is because they ususally failed --- the planes didn't blow up, the stab victims didn't die.
People have strange ideas of how "citizenship" works. The American government has no arrest powers in Shabwa and Anyan Province Yemen. There is no due process in that part of the world whatsoever. The USG had three options in al-Awlaki's case: (1) ignore him and hope that he didn't eventually manage actually blow up a passenger plane, (2) kill him with a targeted airstrike, or (3) invade Yemen to capture him. The USG chose the option that killed the fewest innocent civilians.