Push bike is a particularly English term for pedal powered bikes without engines. I think the term "push bike" has pretty much died out in the UK now. Years ago it was common.
Back then I remember it was common for people to add an engine to a push bike.
This one was common in England, I remember.
The French were keen on designs that fitted in a replacement back wheel. I can't find a picture of the one that was very common and actually worked well. I saw one a few years back on a visit to the UK and I wish I had taken a picture.
Here is one example, of the English Cyclemaster. The French one was more popular.