Some years ago I was a live-in cat sitter for a friend who'd gone on holiday for a couple of weeks. During one of the update phone calls made, I commented that "The cats were missing you". I don't think the owner heard the last word, for I was given the 4th degree interrogation about how they went missing!!
I very carefully had to explain that they weren't missing, as in not here, just missing their owner. I don't think I'd have survived if they really had gone walkabout! If I pet sit again, I'd now choose my words carefully when the owner asks how they're getting on...
But, irrespective of that, I think that being honest about any accident or similar, is the best way to deal with the situation. After all, they did trust you with their pet, so why betray that trust, however difficult it may seem at the time.