Well, without getting into any debate over whether or not psychic powers are real or not, it sounds like you at least want to test to make sure.
The secret is to have a test that proves it without anyone being able to influence the test. The term you want for proof is ‘Independant Double-Blind Study’. By Independant, we mean that the test is conducted by people who have no vested interest in the outcome. By Double-Blind, we mean that neither the tester nor the testee knows what the results are until all the data is collected.
Here is an example of such a test. Suppose your power is to determine if someone is thinking of a wombat.
The subject is seated at a chair at a table across from you. They know that a psychic test of some sort is being performed, but they do not know what is being tested for. Behind a small screen, out of your sight, is a small deck of cards. Each card has a picture of an animal on it (or other objects - chair, bicycle, Van Gogh’s Starry Night, etc). The person conducting the test has randomly shuffled the deck, and has no idea what order they are in. For additional obfuscation, the deck is cut, and only half of the cards are used.The subject looks at the top card in the deck, thinks about the subject for a determined amount of time, then places it in another pile. You record on a list of numbers whether or not you thought they were thinking of a wombat at that time. When the deck is depleted, the tester then examines the discards, and records which cards were drawn and in what order. This is repeated multiple times, with multiple subjects. When a sufficient number of subjects have been tested (more is better!), then the results are compared with the predictions.
Of course, this is a simplified method, but should suffice for preliminary testing.