You could use DATEADD to add 26 years to 1990:
CREATE TABLE #tableName(receivedDate DATE);
INSERT INTO #tableName(receivedDate) VALUES ('1990-01-06'),('1990-01-16');
UPDATE #tableName
SET receivedDate = DATEADD(YEAR, 26, receivedDate)
WHERE DATEPART(yyyy, receivedDate) = '1990' 
  AND DATEPART(mm, receivedDate) = '01';
SELECT *
FROM #tableName;
LiveDemo
SQL Server 2012+ you could use DATEFROMPARTS:
UPDATE #tableName
SET receivedDate = DATEFROMPARTS(2016,
                                 DATEPART(mm, receivedDate),
                                 DATEPART(dd, receivedDate)
                                )
WHERE DATEPART(yyyy, receivedDate) = '1990' 
  AND DATEPART(mm, receivedDate) = '01';
LiveDemo2
EDIT
Keep in that expression  DATEPART(yyyy, receivedDate) = '1990' is not SARGable and query optimizer will skip index ob receivedDate(if exists any). You could use BETWEEN or >= < like Gordon Linoff proposed.