traveller wrote:

If you are travelling on any of our services without either:

• a ticket that is valid and available for the journey you are making
• an Oyster card containing a valid season ticket
• an Oyster card, when you are paying as you go, showing a record of
the start of your trip
[...]
AND we believe that you are trying to avoid paying the correct fare,
you may be prosecuted.

So in addition to travelling without the appropriate ticket it seems
that it is also neccessary to establish a that the passenger is 'trying
to avoid paying the correct fare'. Surely this involves some proof that
the passenger hasn't simply made a mistake?

No. Under the statute TfL uses to prosecute bus fare non-payment, no
such proof is necessary. They only have to prove that you did not pay
the correct fare. What you are quoting is their policy that they will
only prosecute you if they think that you are trying to avoid the
correct fare. But they do not need to prove what they think to make the
prosecution stick.
--
Michael Hoffman