You can use a subquery to check for duplicates. SQL Server supports a delete t1 from table1 as t1 syntax where you can assign an alias to the table you want to delete from:

delete  sc1
from    ServiceCodes sc1
where   sc1.NewlyInserted = 1
        and exists
        (
        select  *
        from    ServiceCodes sc2
        where   sc2.NewlyInserted = 0
                and sc1.ServiceId = sc2.ServiceId
                and sc1.CustId = sc2.CustId
        )

Example at DBFiddle.

A solution that uses the ROW_NUMBER() window function to search for duplicates.

with
  t as (
    select
      newlyinserted,
      row_number() over(partition by ServiceId, CustId
                        order by newlyinserted) as rn
    from ServiceCodes
  )
delete from t
where newlyinserted = 1 and rn > 1;

Try it on db<>fiddle .

Update:
If you need to "delete" and "update", then you need the MERGE statement (all about OUTPUT clause).

with
  t as (
    select
      *,
      row_number() over(partition by CustId, ServiceId
                        order by newlyinserted) as rn
    from ServiceCodes
  )
merge into t
using (values('')) as d(d)
  on newlyinserted = 1
when matched and rn = 1 then update
  set newlyinserted = 0
when matched and rn > 1 then delete
OUTPUT $action, deleted.*;

Try it on db<>fiddle .

UPDATE @tempservicecodes
    SET isNeedstobeDelete = 1
    FROM (
        SELECT 
            MainTempServices.Client,MainTempServices.CompID,MainTempServices.ServiceCode   
        FROM @tempservicecodes MainTempServices  
        LEFT JOIN @tempservicecodes ReqTempServices 
            ON ReqTempServices.NewlyInserted = 0 
            AND MainTempServices.Client= ReqTempServices.Client
            AND MainTempServices.ServiceCode = ReqTempServices.ServiceCode  
        WHERE 
            MainTempServices.NewlyInserted = 1  
            AND ISNULL(ReqTempServices.ServiceCode, 0) = 0
    ) AS ApplicableServices  
    WHERE  
        NewlyInserted = 1  
        AND [@tempservicecodes].Client = ApplicableServices.Client  
        AND [@tempservicecodes].ServiceCode = ApplicableServices.ServiceCode   
        AND [@tempservicecodes].CompID = ApplicableServices.CompID

Delete from @tempservicecodes where isNeedstobeDelete = 1

Somehow, I Found this Left JOIN It is working Fine and Performance also Pretty Good.