I used to be able to access the mashes by clicking on "quiz" then scrolling down to index and then scrolling to "today's mashes". Ten were displayed and then the next hour 10 more were displayed.
Yesterday, I finally found a mash under each quiz type. I played one and was told no score would be recorded as I'd already played that day's mash!!
I played a different theme and was told the same thing.
Finally, I found a theme that actually scored for me.
How do I know whether I've played a mash? How can I ever hope to progress toward the "master masher" badge?

I don't think those are the ones being referenced. I believe they changed the name on you! What you need for this badge is to go to any category, sub-category, etc. and play the "mixed quiz game". Since the changes, there may no longer be a way to see if you've played it that day or not, wasn't one I could find since "recent scores" doesn't seem to show up until after you've played it. But there are literally hundreds from which to choose, so you'll probably be able to remember on a daily basis.

Those mixed quizzes are top-level, from the entire category. For Master Masher, you need to go to the specific subcategory (after selecting the main category, select one of the subcategories displayed) and play the mixed quiz it offers. Since you have to move between subcategories to play each new one, you should be able to keep track of what you have done that day, especially if you are systematic about it.

Thank you both for the responses.
It all changed suddenly for me as I followed my listed procedure on Monday and Tuesday it was gone.
Onward, I guess.

You should find it so much easier now! As looney_tunes mentioned, all those subcategories, just take them in order. That's what I did. Say you like Literature, you've got all those favorite authors and any with at least five (I think that's the correct lower limit) will be a subcategory and prolific authors even have subcategories there. That was one of my major ones and I could just go down the line of my favorites. Works that way for Music, Movies and many other categories.