What you need to do is go into catch-up and on-demand programmes and find programmes with no expiry date. What you need to do is watch one second of each to make your Continue Watching list incredibly long. I can't remember what programmes I managed to do this with, I think the likes of Channel 4 and Channel 5 have tons of those 999 emergency services programmes. Anything like that. Just watch a ton to make the list as long as possible.
A few days after making my list incredibly long it appears that all of the programmes that were on it (including the ones I used to make it long) had all wiped off and the list was empty. This appears to be the only fix available at the moment.
I still cannot believe that the Clear button doesn't work on the Continue Watching screen, absolutely bizarre and a very lazy bit of programming and engineering to say the least.