Before loading the #Paid table, cluster it on all the GROUP BY columns:

Yr
,YrMnth
,Market
,AType
,Groups
,Code

To answer the original q, yes, there is a very easy way to "tell" the trigger not to do the INSERTs into table B.

Use specific byte(s) of CONTEXT_INFO...

Once you've sent a 9-digit number to them, and they've validated it, you shouldn't need to send them that exact same number again.  That is, keep a cache of the...

Do you actually have NULL employee ids?  That's rather bizarre.

If you do, use Thom's code.  If not, all you need to write is:
where employee.cgId = @employeeid

If...

You don't state what issue happens when you add the WHERE.  [You don't need the N' and shouldn't use it.]

Perhaps there's some other character in the column besides...

We could give a more specific answer if you'd show the actual code.
But, for example, if it's something like this:

WHERE ISNULL(date_column, '19000101') >= '20170101'
then you don't...

INSERT INTO dbo.Tbl3 ( ID1, ID2 )
SELECT (SELECT ID1 FROM dbo.Tbl1 WHERE Name1 = @Name1),
    (SELECT ID2 FROM dbo.Tbl2 WHERE Name2 = @Name2)

SELECT Total_Input, Total_OK, Total_Fail,
     CASE WHEN Total_Fail > 0 AND Total_Fail > Total_OK THEN Total_OK
     WHEN Total_Fail > 0 THEN Total_Fail
   ...

SELECT b.time * 1.0 credit, 0.0 credits, b.Id, b.date
FROM staff b
INNER JOIN (
    SELECT *, isnull(dbo.ConvDate2(e.Startdate),'20111101') AS Startdate_converted
    FROM emp