Okay, after a bit of testing, I've found a way to reproduce.
If I click and drag each clue slowly from board to house, I have no issues. But if I rapidly click and/or click drag a clue without placing it onto the house board before releasing click (essentially if I'm playing too quick and/or mis-place the clue onto the wrong spot) then I'm locked out of interacting with clues. A reset fixes normally fixes this, though occasionally it doesn't and I have to hit reset a few times. I can also get locked out by dragging from the clue-description section instead of the header. This is a much more consistent method to lock but also more consistently fixed on reset.