We're back to "it's expensive to be poor". I have friends in the same situation, except they rent their home, so they couldn't even invest in a heat pump if they had the money. Overall they could have a rather significant amount of money each year, if they bought a home. That would require being able to afford the down payment... Which they could, if they had access to the potential savings from owning their own home.

It's a vicious circle, trapping the poorest in a circle of poverty from which there is little escape, even in social welfare states.