Try this...
-- This CTE search for the linked client(child)--
;WITH pr_linked(CLIENT_ID, NAME, LINK_CLIENT_ID, [LEVEL])
AS
(
SELECT CLIENT_ID, Name, LINK_CLIENT_ID, 1
FROM #PR_LINK_INV_HST
--WHERE LINK_CLIENT_ID = @CLIENT_ID
WHERE CLIENT_ID = @CLIENT_ID
For fast, accurate and documented assistance in answering your questions, please read this article.
Understanding and using APPLY, (I) and (II) Paul White
Hidden RBAR: Triangular Joins / The "Numbers" or "Tally" Table: What it is and how it replaces a loop Jeff Moden