Your design is giving you problems.  Be aware you will deservedly get called out for it.

You cannot update to solve your problem.  You need to insert rows into tblA.

insert into tblA 
(DrawingNo
, PartNo
, TestName
, Description
)
select
a.DrawingNo
, a.PartNo
, b.TestName
, b.Description
FROM 
tblA a
inner join tblB b
on a.DrawingNo = b.DrawingNo

You may need to delete all of the rows where the TestName and Description are blank (Null or '')

The results you're looking for are produced by joining tblA to tblB on the DrawingNo field. Since this result produces a different number of rows than tblA you don't want to update tblA. Rather, you want to insert the result of joining tblA with tblB into a new table. SELECT A.DrawingNo , A.PartNo , B.TestName , B.Description INTO #tmpTbl3 FROM #tmpTestTblStage2 AS A JOIN testTblSP AS B ON A.DrawingNo = B.DrawingNo SELECT * FROM #tmpTbl3<

• Edited by PeterDNCO 7 hours 20 minutes ago
• Proposed as answer by Naomi NModerator 3 hours 53 minutes ago

Please follow basic Netiquette and post the DDL we need to answer this. Follow industry and ANSI/ISO standards in your data. You should follow ISO-11179 rules for naming data elements. You should follow ISO-8601 rules for displaying temporal data (https://xkcd.com/1179/).We need to know the data types, keys and constraints on the table. Avoid dialect in favor of ANSI/ISO Standard SQL. 

The results you're looking for are produced by joining tblA to tblB on the DrawingNo field. Since this result produces a different number of rows than tblA you don't want to update tblA. Rather, you want to insert the result of joining tblA with tblB into a new table. SELECT A.DrawingNo , A.PartNo , B.TestName , B.Description INTO #tmpTbl3 FROM #tmpTestTblStage2 AS A JOIN testTblSP AS B ON A.DrawingNo = B.DrawingNo SELECT * FROM #tmpTbl3<

• Edited by PeterDNCO Saturday, September 05, 2015 11:45 PM
• Proposed as answer by Naomi NModerator Sunday, September 06, 2015 3:12 AM

Great response - thank you!