I wouldn’t recommend les gens pauvres, it somewhat looks like child speak to me.

Les pauvres gens is very idiomatic but might have an stronger meaning than just “poor people”. It is not just the fact the people lack money but they are in poor situation. Pauvre gens also convey compassion.

Les personnes pauvres might work but I wouldn’t use it in that sentence.

Les pauvres is right but might be avoided for being too direct. Here, pauvres means people with low financial resources.

La politique du gouvernement vise à aider les pauvres en province.

The politically correct expressions that would be used in such a sentence would be:

La politique du gouvernement vise à aider les personnes en situation de pauvreté résidant en province.

or

La politique du gouvernement vise à aider les personnes démunies/défavorisées en province.

Note that en province is also becoming politically incorrect when it means in France excluding Paris area. Nowadays, you’ll often hear instead: dans les régions.