English Translation
He had an absent look on his face.
放心した顔つき sounds like an unnatural robotic translation. I would propose:
彼は上の空だった。（かれは うわのそら だった）。
彼はぼんやりとした表情を浮かべていた。（かれは ぼんやりとした ひょうじょう を うかべていた。）
放心（ほうしん）per se is a frequent expression, but it’s a bad collocation with 顔つき (countenance).
上の空 is a state of mind when a person is walking in the sky or on the cloud. And the state of 上の空 mind is obviously observed from their facial expression and behavior. So, you don’t need to additionally say 顔つき or 表情 when you use 上の空.
ぼんやりとした表情を浮かべていた is another frequent expression. ぼんやり means “foggy”, “vague”, or “idle”. The person of ぼんやり cannot focus on something. 表情を浮かべる is a better collocation that 表情をする. The connotation of 浮かべる is more passive than する. He was not intentionally with an absent look on his face. So, 浮かべる is better.