Delete a row on a split table

Help! I need to delete some rows in a table that have been split. When I try to delete it deletes the whole row. I don't need it to delete the whole row. How do i only delete one potion of it? Here is an example. I need to get rid of the blank rows. Also I have tried adding rows to one side and it adds a row all the way across. How do I also add a row to one side as well? 

Any help would be greatly appreciated. 

Thanks