I also have the same issue. I have a Samsung A5 at Android 8.0. I didn't get any notification that 8.1 was required but over the last couple of weeks I've had three separate failed payments due to the app just hanging after I enter my passcode. I sent a message to Tesco Bank on 11th May but have not had a response.
I've now uninstalled the app as this is proving to be a real source of frustration and would rather use a OTP, which is quicker anyway. I missed out on theatre seats I'd selected as they were only held for a few minutes while I was waiting for the app to respond, which it never did.
I'm all for extra security but if it's impacting payments then they're doing it wrong - backward compatibility is paramount and every major software vendor builds that in, at least for a reasonable amount of time.
The suggestion of calling a vendor directly is also utter nonsense and a head in the sand approach to an issue that is Tesco Bank's responsibility. The vendor in my latest case was Amazon - there is no way you could talk to anyone on the phone about authorising a payment.
At the very least they should allow the option of selecting a OTP to be sent at the time of payment, as plenty of other banks do.