Engine backfires and then cuts out

I would suspect a clogged catalytic converter. The easiest way to test for that would be to take out the upstream oxygen sensors (the ones closest to the engine) and then see if it runs better. It will be loud, but if it runs better, then the catalytic converter is your problem. The reason that this works is because a clogged catalytic converter creates a lot of back pressure, keeping the engine from breathing out properly, and basically choking it. Removing the sensors allows the engine to breathe as it needs to. Here are some guides to help you:

- How to replace an O2 sensor - https://www.2carpros.com/articles/how-to-replace-an-oxygen-sensor

- How to test a cat - https://www.2carpros.com/articles/how-to-test-a-catalytic-converter

- Symptoms of a bad cat - https://www.2carpros.com/articles/bad-catalytic-converter-symptoms

Let us know what you find and we will go from there.