How can I make an SFX cut that I can actually cut into and drip blood from? I have scar wax, fake blood and skin-tone makeup.

On film, this is done with a latex prosthetic. You make a blood bag, cover it with latex, and make it up to resemble the skin. Then you cut it with an actual knife.
That’s much too dangerous to do on stage: never have a sharp knife on stage.
No, I’m not kidding around there. That’s not “Oh, they say never do it, but really we do it all the time.” Never . People get badly hurt. On film, people are doing exactly one thing, and an expert is right there guaranteeing that everything is being done precisely as they said to do it. On stage it’s just actors, doing a scene, and there is absolutely zero way to do anything safely with a sharp knife. Never, ever, ever.
On stage, you use a trick knife like this . Or you use a blood pack which an actor bursts with their hand, not a knife. It’s usually concealed under clothing, so it can’t show bare skin.
With an enormous amount of effort you can have a blood reservoir concealed on the actor’s body, with a tube leading to an exposed part of the body (like a neck or wrist), covered with latex. Then you “cut” it with a blunt stage weapon, and the actor operates the pump to make the blood. Doing that every night is a real pain in the butt.

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