In fiction, often psychic powers are portrayed as extreme, ultra-powerful abilities. But even very modest powers could provide a profound advantage.
Take psychokinesis, the power to physically manipulate things with your mind. Suppose that your psychokinetic power is very weak. You have a range of just ten centimeters. You have a strength and coordination comparable to that of a single hand of first grader.
What can you do with that?
You can open locked doors that don't require a key to unlock from the inside. You can shock someone in your close proximity by socking them in the nose or ears or poking them in their eyes. You can trigger the car alarm on someone else's keys when you are near them, or pick pocket their keys or wallet. You can retrieve jewelry that has fallen into a drain or a sink. You can turn cut the power to a car by pressing a start button or turning a key in the ignition from the passenger's seat. You can zip up or button the back of your dress that you can't reach with your hands. You could pull the trigger on a concealed firearm without moving your hands towards it. You could create a distraction by opening a faucet.
Take telepathy, who can communicate silently with their mind. Suppose that you are a very weak telepath. You have to be touching someone for it to work. You can't communicate outright words or images, only vague urges or strong feelings.
What can you do with that?
You could easily determine if someone is living or brain dead, or if they are human or a robot. You could tell if they were seething with rage inside despite having a calm and civil demeanor. You could manage your relationships better. You could help someone understand you when you can't speak even if you can't be very specific.
Consider pyrokinesis, the power to create or manipulate fire with your mind. You are weak. You have a range of perhaps ten centimeters, and can only make a little spark.
What can you do with that?
Sure, you can light a cigarette or a candle or gas stove, when your matches are wet and you don't have a lighter. But you could also trigger pretty much any fuel tank to explode. You could start a wildfire in the vicinity of flammable materials like dry grass. You could squelch the trigger to a stick of dynamite. You could use it to signal a coded message as you flicked it on and off. You could burn the inside of someone's mouth, or singe or ignite their greasy hair, or create the equivalent of a cigarette burn in some unexpected place to distract them so they'd unhand you.
Consider the ability to see into the future, but very weak. Only within the range of your senses and only for a few seconds in advance.
What can you do with that?
This is essentially the main edge that the Jedi have in Star Wars, allowing them to be just a little bit ahead of people that are fighting them. It could allow you do avoid all manner of accidents. It could prevent you from being betrayed by someone at the last minute, or just prevent you from doing something embarrassing. It might allow you to defuse a bomb by foreseeing what happens if you cut the red wire. It could be very useful on game shows. You could win every time at rock-paper-scissors, or could allow you to know whether to call heads or tails in a coin flip.
You can imagine similar ways that other psychic powers, even if extremely weak, could be game changing.