Would adding an Index on the AddedDate column help minimize the Sort 70.4% impact
The table dbo.tblhall_contacts_ajax_search_lkp LKP has a surrogate PK and no other indexes. Wondering adding an index on the ChildID and ParentID columns on LKP would also help.
Are those columns 22 and 23 in table 7? If so, then yes, an index on those two columns (either ix(col22,col23) or ix(col22) include(col23)) should allow them to be pulled from the index without an additional sort operator. That said, these columns are ultimately pulled from a table scan – is there a reason there is no clustered index on this table?
What other changes that may make to the plan (I didn't inspect where else that index might be used, if it exists) and of course the impact that will have on the DML portion of your workload is a different issue altogether.