Judge Richard Rundell said serious delays stopped Child B making contact with his extended family.
The judge said it was appropriate compensation for the "lost opportunity over three years to develop a relationship with at least one of his half-siblings".
The case was at Worcester Family Court.
A care order was made for B in February 2011, and Worcester County Council was ordered to withhold contact with his mother following concerns that she could have a potentially "harmful impact" on him.
The man thought to be his father had refused to take a DNA test and played no part in the case.
The court heard the council initially decided that B, who has behavioural difficulties, should be adopted.
When no appropriate adopters were found the plan was changed to allow long-term foster care in 2012.
The judge said the original adoption placement order should have been revoked and the matter returned to court, with members of his family given a fresh chance to contact him.
But that did not happen for three years, until February 2015.
During the intervening period there were seven "looked-after child" (LAC) reviews and a failure by an independent reviewing officer (IRO) to take adequate action.
The judge said both the local authority and the IRO now conceded they had breached B's rights under Article 6 of the European Convention on Human Rights (ECHR) to a fair hearing, and his Article 8 right to "private and family life".
Both the authority and IRO have apologised for failing for three years to "take the obvious step" and say there would be "no repetition of such a lengthy failure", Judge Rundell said.
A 10-year-old boy has been awarded £5,000 after a judge ruled his human rights had been breached while in local authority care.