Child Sexual abuse - can you jail person just based on what child
Under U.S. law, yes.
There is no proof whatsoever about this, no facts, or witnesses
Testimony of a witness is evidence.
There is a witness, the child, and another witness, the psychologist, who heard the child and testified. Those are facts and witnesses.
Assuming that an exception to the hearsay rule and confrontation clause can be secured (which it certainly can for purposes of a probable cause determination to make an arrest and sometimes can and sometimes can't for purposes of a conviction), what the child said can be presented through the psychologist, which is admissible evidence.
The fact that something was said by a child does not make it not evidence, nor does it make it inadmissible. The rules governing when hearsay is admissible as evidence are arcane at best and very circumstance dependent.
A jury can choose to believe that any particular item of evidence is sufficiently credible to constitute proof beyond a reasonable doubt. Credibility and weight are for a jury, not for the judge and not for an appellate court.