Therefore, you are allowed to apply for characters with a certain degree of psychic/psionic talent. However, keep in mind that this is not a commonly acknowledged or trained skill in Pernese society. It is simply, superstitiously known that "some people have power," and it's generally associated with dragonriders. These are the people, in fact, that are Searched -- the ones most likely to Impress are the ones who already have a certain amount of telepathic or tele-empathic ability, and who are therefore most readily able to accept the Impression bond. It is not terribly uncommon, then, to have some small scrap of power -- the key phrase being 'some small scrap'. Very few people are going to have even Lessa's level of ability, and consider that she is extremely powerful by Pernese standards. Think subtle. There will be no boulders being flung around by the power of some human's mind alone, no pyro- or cryokinesis (at least for now ... *evilwink*) and nothing else ostentatious or clearly disruptive to the setting. Very minor telekinetic talent (levitation-type power just strong enough to make yourself or an object a bit lighter or heavier rather than lift you/it all the way off the ground; the ability to 'nudge' the course of a thrown or fired weapon or object to improve your aim and make it more likely to hit its target; et cetera) is acceptable, but most people with power will have telepathic or tele-empathic talents (they will sometimes, rarely at will [albeit with effort] be able to hear what other people are thinking, or feel what they're feeling.)
As mentioned, you are allowed to apply for a character who has some sort of power. This is especially likely if you're applying for a dragonrider or a Candidate, as those people in the right age group are far more likely to be Searched (in fact, people with no empathic ability are never Searched, as those people are physically and emotionally incapable of making Impression and the Search dragons sense it.) That said, please read this section thoroughly first and think very hard about how this will affect your character's life before being Searched. The average Pernese is not necessarily as understanding or positive about this kind of 'power' as the Weyrs. They probably won't be exposed to outright superstitious oppression or violence, but they might well suffer from social awkwardness and might be subtlely ostracized. At minimum, they will not be easily understood by those around them. Being Searched might well come as a relief!