Definition: to continue doing something with a lot of energy, often because it is important or because there is not a lot of time to do it.
(This phrasal verb has more than one meaning)
E.g.1: The weather is getting bad – we should push on if we want to home before it gets dark.
E.g.2: I should push on with this work – the boss wants it by 3 p.m.
This phrasal verb cannot be separated.