Harry's central motive in marrying Moo is to take her money from her, which he does. He and Moo's sister appear to have an unholy alliance and appear to shoot Moo accidentally in the head. The "accident" is called into question when Harry puts Moo unceremoniously in an insane asylum, a common repository for unwanted wives in the 19th century, and flees with her money in hand.
At the start of Moo by Sally Clark, I don't get the sense the Harry shoots Moo on purpose. Both Moo and Harry are familiar with, and comfortable around, guns. While Moo can hit a can thrown in the air, Harry can hit the can and keep it in the air by hitting it repeatedly with bullets from his rifle. Moo has just kiddingly told Sarah that she will marry the first man she meets who is better with a gun than she is. After witnessing Harry's shooting, as he casually walks away, stage direction tells us:
MOO stares after him. She throws the can up, tries to shoot it, misses. She stares back after HARRY.
Here is a man who is better with a gun than she is.
By the time Harry shoots Moo the first time, he has been accepted by Moo's parents (strangely enough) as a replacement for Moo's dead brother, George, who has been killed in the war. During this scene, Ditty (Moo's sister) who has had, it would appear, no interest in learning to shoot, is having Harry show her how. Ditty is shamelessly flirting with Harry. While she might not shoot a gun, we can assume that she at least knows the basic parts of a gun, and how one aims and shoots. Still, she adopts a "helpless female" attitude:
And you—um—put your hand there. Now, don't pull the trigger.
Yeah, that thing there [...] now you look down the barrel.
And while it is obvious as to what Ditty is doing, it doesn't take long to figure out that Harry is aware as well. As he shows her the sight on the gun, his face is very close to Ditty's; and as he instructs her in taking aim, he kisses her on the neck. Ditty, startled, cries out, the gun goes off, and both Ditty and Harry are thrown to the ground. Harry seems much more occupied with kissing Ditty than aiming the gun anywhere, let alone at Moo. When the two sit up, they look over to Moo and see that her head is bleeding badly. Ditty claims (erroneously) that Harry has killed Moo (an example of foreshadowing).
There is nothing in the stage direction that indicates that Harry purposely tried to harm Moo. As a rotter (as the author calls him), his goal is to seduce women, take their money and then abandon them. Killing Moo would be counterproductive to his scheme: and it is, after all, Moo that is willing to leave everything behind and run off with Harry. Inferring that he has a sense of her interest, killing her seems out of the question. I think it is an accident.