To have task 25 start 10 days after task 24 ends, you can either:
- Make task 25 a 10-day duration (and therefore will show a due-date of 10 days after task 24's due date) and list 24 as the predecessor
- Or if you don't want it to look like task 25 takes 10 days, have its duration be 0days and put 24fs+10d as the predecessor (10 days after task 24).
In terms of not showing in the My Task widget at all, this sounds like an odd request but the user could just add a filter on her side to only show tasks due [this week] for example or only those that 'can start.' Or can use a grouping instead to group by week due.
If this helped you, please mark correct to help others : )