That's a good reason to be worried about what you're doing. You will have this issue whether you're using GoAnywhere or any other product though.
Your key factor here is identifying which records were pulled (transferred to the other database). You'll need to figure out some sort of functionality to identify these records for the Delete statement. This could be making a temp table of the ones you're transferring, then stating a "Delete from table where record exists in temp.table", then when all records have been transferred, then deleted .. remove the Temp.Table.
If they are date/time stamped, delete where records are older than the last time of transfer. Delete by record number. Or, transfer them record by record deleting after transfer.
The key here is knowing how to identify the record(s) that have been transferred so that you can delete them specifically, not a "Delete * from <table>".
Hope this makes sense.