It sounds like your lagging, Sometime, no a lot of the times when I d/c for a few seconds it connects back and plays their card at 1 sec left and when its my turn I have like 5 secs to play. I mean this may not be it but this happens to me wayyyy to often
I face this too. Usually when there is 20-30 secs left for the opponent to play i send out a Kate emote.
If the opponent has already played and its not showing up, Kate emote fixes this by triggering the opponent action 1st then the emote.
If the emote just pops up with out anything else, that means the opponent has not played his/her turn.
If the Kate emote doesn't pop up, refresh/restart whatever client your are playing on.