|=TODAY()+10 does not add 10 days to the date in A1.|
It adds 10 days to the current day's date regardless of what is in any cell anywhere in the spreadsheet. =TODAY() is a standalone function that is not dependent on any other cell.
That said, you can decide whether it displays or not by using an IF function to check if A1 is populated, just as long as you are aware that regardless of what date you put in A1, you will always get the current day's date plus 10 days when you use =TODAY()+10.
=IF(A1<>"", =TODAY()+10, "")
=IF(A1<>"", TODAY()+10, "")
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