Cardboards for puppet's mouth.  Now the boards are breaking apart.  How can I fix the cardboards?  Right now, I am sticking Duct tape in the cardboard.   

If you can flip the puppet's head inside out or get to it well enough you could glue either fabric, craft foam or leather over the cardboard.

...and if you can't turn them inside out...cut another mouth plate out of stiff card , pre-glue the card, and carefully insert the mouth plate and glue it to the  existing mouth plate. As Shawn said, you can cover the non glued side of the new mouth plate with some fabric craft foam or leather, to get more life out of the repair. If I have to do this, I usually glue a 'sock type' liner to the new mouth plate before inserting it. So that the foams wear less, when inserting your hand hundreds of times... ( I have this situation coming up on a pig I am using at the moment. I was too lazy, or too rushed, to do a lining when I put the pig together, Now I am regretting that haste.)

Thank you both.  What I did is cut up craft foam, glue it and stuck it into my two puppets.  Worked.  Thank you again.   

Here is an example of adding a liner and mouth card after the fact. I designed very poorly yesterday, so I redesigned and built this today, and because of yesterday's disaster, I did not bother with putting a liner inside, until I had the puppet finished. So, I did the 'light card mouth plate to cloth liner' way, and pre-glued and inserted the liner with its light  mouthplate into, or onto the  mouth plate already in the puppet. Done.

Some one told me long ago to use pressed cereal box type cardboard and a couple of layers glued together with white glue and put a big book on top of it so it stays flat as it dries.  That pressed cardboard is sturdier than poster board or corrugated board but still light. It works for small hand puppet but not stiff enough for larger puppets. Glad you got it working again.