IOS – Remove completed reminders in OS X and iOS

On the iPad Reminders app, there is no + button when viewing Completed tasks.

What happens when you click on Data? Can you then add items?

Edit

It looks like others are having the same problem. Does this reply work for you:

here's the solution that seems to work... have tried it, reversed it, and retried and it worked twice.  Turn on reminders in icloud.  go to reminder app.... + should now be there.  i then turned off reminders becasue it seems to just drain the battery... but select option to keep remdiners on the iphone when it prompts you.

Quoting a solution I found in the Apple community web site:

I was having this problem. I changed the syncing range on the phone (under Settings > Reminders) to sync All Reminders. I then logged into iCloud and opened the Reminders tab. This actually caused the reminders on the phone to match the ones in iCloud. I then made changes on my phone while that tab was still opened and the changes started to appear in iCould (sometimes takes a few seconds - up to 20 or 30) and synched on my iPad as well. After that, all of my changes started to sync across all devices - iPhone, iPad,Mac and in iCloud, regardless of where I made the initial change.

Hopefully this will work for you too.

Link here.