So, what happens if the Bus card-reader is broken, and the driver
waves you on - happens a couple of times per month on the routes that
I use (308, 678). Take it I should insist on paying cash?

The official TfL position is that you should pay the cash fare and request
a refund of the price difference from them...

I actually asked about broken readers on buses on "Ask Oyster" in June last
year. The response was...

"In response to your query, in the situation that the Oyster reader is not
working, generally the bus driver will wave you on. In that situation they
should log that their reader is faulty, so if a Revenue Inspector gets on
they will understand the situation. Revenue Inspectors also carry hand held
Oyster readers so if they do scan your Oyster card and see that it is Pre
Pay they will be able to deduct the appropriate fare from your card.

"If you are in a situation where the driver does make you pay, unfortunately
you will be required to pay the cash fare in order to travel. If you do
encounter this situation and have to pay the £1, we can refund the thirty
pence to you if you fill in a Refund of Fares form and send us the proof of
payment."

Maybe this isn't the answer TfL would 'officially' approve of, but it is the
one they gave me.

G.