I currently don't have many good tips to offer since I'm practicing writing this genre myself (Thus I would have no idea if I'm successful or not since my work has yet to be shown to anyone), but I guess one of the most important things to do (IMO) is avoid adding anything supernatural.
Psychological horrors are scary because the nature of the horror in this genre not only makes the audience feel uncomfortable, but they seem like things that can actually happen in real life. Unless you're a talented writer who can convince the audience otherwise (Like Satoshi Kon), supernatural and psychological horror just don't mix well. When you throw in ghosts and super powers, it really lessens the impact of how believable the overall story can be. At most, they should only be hallucinations and whatnot, but then you would have to give a reasonable explanation as to why people were seeing them.
How you write a psychological horror is really up to you since the genre comes in three different forms (That I know of). There's Mind torture, Fridge Horror, and Mind tripping.
Mind torture, as its name implies, is when the protagonist is being mentally tortured. It can be anything like reliving the biggest mistake of their life in their head, or having the villain prey on their minds by making them recall a traumatic memory. This is sometimes categorized as psychological thriller as well depending on how it's used. Death Note is an example of the thriller aspect.
Fridge Horror, from what I can understand, is when the horror behind a certain situation doesn't occur to the audience until much later when they actually think about it. I will provide a good example from The Legend of Zelda: Majora's Mask since it contains so many of these.
Throughout the whole game, you get three masks that help you transform into these: A Deku Scrub, a Goron, and a Zora. However, to obtain such masks, the people whom you get them from must die so that their spirit can inhabit the mask and allow you to transform into them. Although the game shows how Link obtains the Goron mask and the Zora's mask, it never revealed where Link's Deku Mask came from. We are just lead to believe that Skull Kid transformed him into a Deku Scrub and he got a mask out of it by playing the Song of Healing.
Later on when you race the Deku Butler to obtain the Pig Mask, he tells you that you remind him of his son whom left somewhere. Once you complete the game and view the ending, you see the Deku Butler bowing down in front of the dead Deku Scrub whom you see at the very beginning of the game when you first enter Termina. Although you probably didn't really think much about the Deku Scrub and its significance upon your encounter with it back then, the Fridge Horror comes in when seeing the Deku Butler there hits you with the sudden realization that the Deku Scrub is his son, and also that the Deku Mask you hold contains his spirit.
And finally, Mind tripping is when you mess with the minds of the audience, whether it be for horror or just for the sake of weirdness.
The film "The Uninvited"
is a good example for horror usage, while Paranoia Agent is a great example of "for the sake of weirdness".
I recommend watching several animes and movies that fall under these categories. They can provide a couple of neat concept and ideas that you can use, as well as showing how they're executed.
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