You're not referencing your CTE on your UPDATE statement. Additionally, you don't have anything to make a relationship (and that's caused by the previous reason).
You don't actually need a CTE, something as simple as a JOIN should work.
UPDATE ce SET
FormerBranch = CASE WHEN e.EntityName IS NULL
THEN ' '
ELSE e.EntityName
END
FROM ClientExtra ce
JOIN Clients c ON ce.ClientRef = c.ClientRef
LEFT
JOIN Entities e ON c.BranchRef = e.EntityRef AND e.Type = 3