In the Hammer House of Horror TV series episode The House That Bled to Death I did not understand why the girl Sophie killed his father? Also how the cat was killed, was there any ghost/sprit of the old women?
There are multiple ways to interpret the ending, and ultimately I don't think we can point to one of them and say "That is the right ending; the other one is just wrong." I think the screenwriter was happy to leave that uncertainty, because horror flourishes better in an atmosphere where things are uncertain.
To start with what's relatively certain... It seems pretty certain that William was the one who killed Sophie's cat. From William and Emma's conversation in the pool, it seems obvious that he was the one who made most of the frightening events happen. If the cat had really died by cutting itself on a broken window (which is what supposedly happened) he might have remarked about what an extraordinary stroke of luck it was, since it was one of the most frightening of the events... but he says no such thing, so the natural conclusion is that it is one more thing that he engineered to bolster the "haunted murder house" story.
Notice that after the animal's death, William refers to it more than once as "the stupid cat" or "a stupid cat", which no one else does - it may be that he's trying to dismiss its importance out of guilt for killing it. Or, it's also possible that he never liked the cat, perhaps because his daughter seems fonder of the cat than of her parents (Emma makes that statement, early in the episode.)
So it seems unlikely that any of the "haunting" events which happen in the house were actually supernatural. If there was any part of those events which was not actually engineered by William, he likely would have mentioned what a good stroke of luck it was.
So, the other question is: is any part of the ending, after they've left the house, supernatural?
Well, it doesn't have to be. When Sophie leaves the pool and goes upstairs, she reads the book that the estate agent wrote about the "hauntings" in the house. She sees the photo of Timmy, the cat she loved, and she flashes back to Timmy's bloody corpse. She then opens one of the suitcases they took from the house, and finds some of the props that were used to frighten the neighbor Jean earlier - the bloodied knitting, and the eyeglasses, which William denied ever seeing.
She now has all the pieces to put together the truth: that every terrifying, traumatizing thing which happened to her in that house, including the death of her beloved pet, all the things that are still giving her nightmares three years later, were all just a hoax perpetrated by her own father.
She might be killing him just for revenge - revenge for her trauma, revenge for her murdered cat. The whole narrative makes sense that way, and there's no need to invoke anything actually supernatural.
But horror and irony go very, very well together, so it's tempting to ask: what if Sophie's murder was prompted by supernatural influence? What if it wasn't just her own anger at her father, but some influence from the house? There's nothing in the episode that proves such an influence, but there's an intriguing line which leads to some speculation: when William sees the distinctive sword in Sophie's hand, and asks "Where did you get that?" he might mean "I had those swords packed away in a place I never expected you to go looking; why did you go poking into that stuff?"
Or, he might have meant "How on Earth did you get those? I thought I got rid of them entirely when we had finished giving Mr. Peters the 'haunted house' story he was selling!" Earlier, the swords were supposedly reappearing by supernatural means, despite having been taken away by the police and later thrown away (supposedly) by William. If they really did come to Sophie's possession by supernatural means, it would be deeply ironic - all the more so because it would be those who tried to profit by exploiting a tragedy, and the supernatural, meeting their own tragedy from the supernatural.
So, ultimately, you can take either interpretation, depending on which you like. There is no "proof" that anything supernatural occurred, but believing that it did adds an extra layer of closure to the scene.